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FRONTISPIECE 




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THE 



PLANETARIUM, 



ASTRONOMICAL CALCULATOR, 



CONTAINING 

The Distances, Diameters, Periodical and Diurnal Revolutions of all the Planets in 

the Solar System, with the Diameters of their Satellites, their distances from, 

and the periods of their Revolutions around, their respective Primaries; 

together with the method of calculating those Distances, Diameters and 

Revolutions. Also, the method of calculating Solar and Lunar 

Eclipses ; being a compilation from various celebrated authors, 

with Notes, Examples and Interrogations ; prepared 

FOR THE USE OF 

SCHOOLS, ACADEMIES, AND PRIVATE LEARNERS. 



BY TOBIAS OSTRANDER, 

TEACHER OF MATHEMATICS, 

AND AUTHOR OF "a COMPLETE SYSTEM; OF MENSURATION," " THE ELEMENT? 

OF NUMBERS," " EASY INSTRUCTER," " MATHEMATICAL EXPOSITOR," &C. 



ct Consult with reason, reason will reply, 
" Each lucid point, which glow3in yonder sky, 
" Informs a system in the boundless space, 
;i And fills with glory its appointed place ; 
"With beams unborrow'd brightens other skies, 
" And worlds, to the« unknown, with heat and life supplies. 



NEW YORK: 
M'ELRATH, BANGS, & HERBERT, 

STEREOTYPED BY J. S. REDFIELD & CO. 

1833. 




Entered according to the Act of Congress, August, 1833, "by 

tATH, BANGS 
District Coui 
New York. 



IVTELRATH, BANGS & CO., 

In the Clerk's Office of the District Court of the Southern District of 



#*$££ 



r 



PREFACE. 



In presenting the following pages to the public, I will briefly re- 
mark, that the people generally are grossly ignorant in the import- 
ant and engaging science of Astronomy. Scarcely one in a county 
is found capable of calculating with exactness and accuracy the 
time of an eclipse, or conjunction and opposition of the Sun and Moon. 
Is it for lack of abilities 1 No.— There are no people in the world 
who possess better natural faculties for acquiring knowledge of every 
description, than those who inhabit the United States of America. 
In this land of liberty, much has been done, and much still remains to 
be done, for the benefit of the rising generation. Schools, Academies, 
and Colleges have been erected, for the purpose of facilitating and 
extending information and instruction among the youth. Gentlemen 
possessing the most profound abilities and acquirements, have en- 
gaged in the truly laudable employment of disseminating a know- 
ledge of the sciences, both of the useful and ornamental description. 
Still, this branch of the Mathematical science, called Astronomy, 
has been almost totally neglected, especially among the common 
people. From what source has this originated 1 I answer, from a 
scarcity of books, well calculated to give the necessary instruction. 
Though there be many productions possessing merit, which are of 
importance to the rising generation, yet they are deficient in the 
tables necessary for the calculation and protraction of eclipses. The 
works of Ferguson, Enfield and others, from which this is princi- 
pally compiled, contain all that is necessary ; but the expense renders 
them beyond the means of many, who perhaps possess the best abili- 
ties in our land. Extensive volumes are not well calculated for 
the use of Schools ; for a student is under the necessity of reading 
so much unessential, and uninteresting matter, that the essence is 
lost in the multiplicity of words ; and for these reasons, many of the 
teachers have neglected this useful and important branch of the 



4 PREFACE. 

Mathematical science. I have long impatiently beheld the evil, 
without an opportunity of providing a remedy, until the present 
period. 

I now present to this enlightened community, a volume within the 
means of almost every person ; containing all the essential parts of 
Astronomy, adapted to the use of Schools and Academies ; made so 
plain and easy to be understood, that a lad of twelve years of age, 
\vh03e knowledge of Arithmetic extends to the single rule of pro- 
portion, can, in the short space of one or two weeks, be taught to 
calculate an eclipse : and many possessing riper years, from the 
precepts and examples given in the work, will be found capable of 
accomplishing it, without the aid of any other teacher. 

The Tables, (with the exception of two,) I have wholly calculated, 
and then duly compared them with those of Ferguson. Great care 
has also been taken, to present the work to the public, free from 
errors. 

Should the following pages meet the approbation of a generous 
and enlightened community, and be the means of extending the 
knowledge of this important branch of Education, not only to the 
rising generation but to those of maturer years, the Compiler, whose 
best abilities have hitherto been employed in endeavouring to melio- 
rate the condition of man, by improving the mind and enlightening 
the understanding, will have the pleasing reflection, that he has 
removed some of the shackles of ignorance, and supplied a fund of 
useful and interesting knowledge. 

THE COMPILER. 



CONTENTS, 



Preface 



Of Astronomy in general 



Of the Solar System 

Transits of Mercury . 

Transits of Venus 

Of Vesta • . 

<3f Juno . 

OnjCeres 

On Pallas 

On Jupiter 

On Saturn 

On Herschel, or Uranus 

On Comets 



On Gravity 



Page. 



SECTION I. 



SECTION II. 



SECTION III. 



SECTION IV. 

Phenomena of the Heavens, as seen from the Earth 



14 
18 

20 
23 
25 
26 
26 
27 
30 
32 
34 



SECTION V. 

Physical causes of the motions of Planets . 

SECTION VI. 
On Light and Air 

SECTION VII. 
To find the diameter of the Earth, and the distances of Planets 
from the Sun, &c. . ■ • . • 6? 

SECTION VIII. 

Of the Equation of Time, and precession of the Equinoxes . 73, 

SECTION IX. 

Of the Moon's Phases i .-;.. : . ... 78 

Of the Phenomena of the Harvest Moon . • .81 

1* 



6 CONTENTS. 

Page. 
SECTION X. 

On Tides ' . 90 

SECTION XI. 

Astronomical Problems . . . . .97 

SECTION XII. 

On Eclipses . . . . . . 106' 

SECTION XIII. 
On the Construction of Astronomical Problems . 118 

SECTION XIV. 

Directions for the Calculation of Eclipses . . 132 

Astronomical Tables ...... 136 

Equation of Time . . . . : . 159 

SECTION XV. 

Examples for the calculation of Eclipses . . 167 

To find the Sun's true place .... 179 

Concerning Eclipses of the Sun and Moon . . . 180 

SECTION XVI. 
To project an Eclipse of the Sun .... 182 

SECTION XVII. 

Projection of Lunar Eclipses ..... 192 

SECTION XVIII. 

On the Fixed Stars ...... 200 

On Groups of Stars . . . . . .203 

On Clusters of Stars . . . . . . 203 

SECTION XIX. 

An account of the Gregorian or New Style . . : 206 

Chronological Problems . . . . 207 

Perpetual Almanack . . . . . .211 

Tabular View of the Solar System . . . . 212 

SECTION XX. 

Problems for the Terrestrial Globe . . . .214 

Problems for the Celestial Globe .... 222 

Dictionary of Astronomical Terms .... 225 



^ 



THE 



PLANETARIUM, 

AND 

ASTRONOMICAL CALCULATOR. 



" Astronomy, Parent of Devotion ! engage my midnight vigils , 
Elevate my thoughts to contemplate thy vast realities ; 
Warm my soul with adoration, pure and fervent praise, 
Tc Him, whose finger fashioned yon revolving worlds !" 



SECTION FIRST. 

OF ASTRONOMY IN GENERAL. 

Of all the sciences cultivated by mankind, Astronomy- 
is acknowledged to be, and undoubtedly is, the most sub- 
lime, the most interesting, and the most useful. By the 
knowledge derived from this science, not only the mag- 
nitude of the earth is discovered, the situation and extent 
of the Countries and Kingdoms ascertained, trade and 
commerce carried on to the remotest parts of the world, 
and the various products of several countries distributed, 
for the health, comfort, and conveniency of its inhabit- 
ants ; but our very faculties are enlarged, with the gran- 
deur of the ideas it conveys, our minds exalted above 
the low contracted prejudices of the vulgar, and our un- 
derstandings clearly convinced, and affected with the 
conviction, of the existence, wisdom, power, goodness, 
immutability, and superintendency of the Supreme 
Being. So that without any hyperbole, every man ac- 
quainted with this science, must exclaim with the im- 
mortal Dr. Young : " An undevout Astronomer is mad." 
From this branch of Mathematical knowledge, we also 
learn by what means, or laws, the Almighty Power and 



8 OP ASTRONOMY IN GENERAL. [SEC. I. 

Wisdom of the Supreme Architect of the Universe, are 
administered in continuing the wonderful harmony, 
order and connexion, observable throughout the plane- 
tary system ; and are led by very powerful arguments, to 
form this pleasing and cheering sentiment, that minds 
capable of such deep researches, not only derive their 
origin from that Adorable Being, but are also incited to 
aspire after a more perfect knowledge of his nature, and 
a more strict conformity to his will. 

By Astronomy we discover, that the earth is at so 
great a distance from the sun, that if seen from thence, it 
would appear no larger than a point ; although its dia- 
meter is known to be nearly 8,000 miles : yet that dis- 
tance is so small, compared with the earth's distance from 
the fixed stars, that if the orbit, in which the earth moves 
round the Sun, were solid, and seen from the nearest star, 
it would likewise appear no larger than a point ; although 
it is at least 190 millions of miles in diameter ; for the 
earth in going round the sun, is 190 millions of miles 
nearer to some of the stars, at one time of the year than at 
another ; and yet their apparent magnitudes, situations, 
and distances, still remain the same ; and a telescope 
which magnifies above 200 times, does not sensibly mag- 
nify them ; which proves them to be at least, one hundred 
thousand times further from us, than we are from the sun. 

It is not to be imagined that all the stars are placed in 
one concave surface, so as to be equally distant from us ; 
but; that they are placed at immense distances from one 
another, through unlimited space, so that there may be 
as great a distance between any two neighbouring stars, 
as between the sun from which we receive our light, and 
those which are nearest to him. Therefore, an observer 
who is nearest any fixed star, will look upon it alone as a 
real sun ; and consider the rest as so many shining points, 
placed at equal distances from him in the firmament. 

By the help of telescopes, we discover thousands of 
stars which are entirely invisible, without the aid of such 
instruments, and the better our glasses are, the more 
become visible. We therefore can set no limits to their 



SEC I.] OF ASTRONOMY IN GENERAL. 9 

numbers, or to their immeasurable distances. The cele- 
brated Huygens carried his thoughts so far, as to believe 
it not impossible, that there may be stars at such incon- 
ceivable distances, that their light has not yet reached 
the earth since their creation ; although the velocity of 
light, be a million of times greater than the velocity of 
a cannon ball at its first discharge ; and as Mr. Addison 
justly observes, " This thought is far from being extra- 
vagant, when we consider that the Universe is the work 
of Infinite Power, prompted by Infinite Goodness, and 
having an Infinite space to exert itself in ; therefore our 
finite imaginations can set no bounds to it." . 

The Sun appears very bright and large, in comparison 
of the fixed stars ; because we constantly keep near the 
Sun, in comparison to our immense distance from them. 
For a spectator placed as near to any star, as we are to 
the Sun, would see that star to be a body as large and 
bright as the Sun appears to us : and a spectator as far 
distant from the Sun, as we are from the stars, would see 
the sun as small as we see a star, divested of all its cir- 
cumvolving planets, and would reckon it one of the 
stars, in numbering them. 

The stars being at such immense distances from the 
Sun, cannot possibly receive from him so strong a light 
as they appear to have, nor any brightness sufficient to 
make them visible to us ; for the Sun's rays must be so 
scattered before they reach such remote objects, that they 
can never be transmitted back to our eyes ; so as to ren- 
der these objects visible by reflection. Therefore the 
stars, like the sun, shine with their own native and un- 
borrowed lustre ; and since each particular one, as well 
as the Sun, is confined to a particular portion of space, it 
is evident that the stars are of the same nature with the 
Sun ; formed of similar materials, and are placed near the 
centres of as many magnificent systems ; have a retinue 
of worlds inhabited by intelligent beings, revolving round 
them as their common centres ; receive the distribution 
of their rays, and are illuminated by their beams ; all of 
which, are lost to us, in immeasurable wilds of ether, 



10 OF ASTRONOMY IN GENERAL. [SEC. I. 

It is not probable that the Almighty, who always acts 
with Infinite Wisdom, and does nothing in vain, should 
create so many glorious suns', fit for so many important 
purposes, and place them at such distances from each 
other without proper objects near enough to be benefitted 
by their influences. Whoever imagines that they were 
created only to give a faint glimmering light to the inha- 
bitants of this globe, must have a very superficial know- 
ledge of Astronomy, and a mean opinion of the Divine 
Wisdom ; since by an infinitely less exertion of creating 
power, the Deity could have given our earth much more 
light by one single additional Moon. 

Instead of our Sun, and our world only in the Uni- 
verse, (as the unskilful in Astronomy may imagine;) 
that science discovers to us, such an inconceivable num- 
ber of Suns, Systems, and Worlds, dispersed through 
boundless space, that if our Sun, with all the planets, 
Moons, and Comets, belonging to the whole Solar Sys- 
tem, were at once annihilated, they would no more be 
missed by an eye that could take in the whole compass 
of Creation, than a grain of sand from the Sea shore ; the 
space they possess, being comparatively so small, that 
their loss would scarcely make a sensible blank in the 
Universe. Although Herschel, the outermost of our 
planets, revolves about the Sun, in an orbit of three thou- 
sand six hundred millions of miles in diameter, and some 
of our Comets make excursions more than ten thousand 
millions of miles beyond his orbit, and yet at that amazing 
distance, they are incomparably nearer the Sun, than 
to any of the fixed stars, as is evident from their keeping 
clear of the attractive power of all the stars, and return- 
ing periodically by virtue of the Sun's attraction. 

From what we know of our own System, it may be 
reasonably concluded that all the rest are with equal wis- 
dom contrived, situated and provided with accommoda- 
tions for the existence of intelligent inhabitants. Let us 
therefore take a survey of the System to which we be- 
long, the only one accessible to us, and from thence we 
shall be better able to judge of the nature and end of 



SEC. I.] OP ASTRONOMY IN GENERAL. 11 

other systems of the Universe. Although there are al- 
most an infinite variety in the parts of Creation, which 
we have opportunities of examining ; yet there is a gene- 
ral analogy running through, and connecting all the 
parts into one scheme, one design of disseminating com- 
fort and happiness to the whole Creation. To an atten- 
tive observer, it will appear highly probable, that the 
planets of our System, together with their attendants, 
called Satellites or Moons, are much of the same nature 
with our earth, and destined for similar purposes ; for they 
are all solid opaque globes, capable of supporting animals 
and vegetables ; some are larger, some less, and one nearly 
the size of our earth. They all circulate round the Sun, 
as the earth does, in a shorter or longer time, according 
to their respective distances from him, and have, (where 
it would not be inconvenient,) regular returns of Summer 
and Winter, Spring and Autumn. They have warmer 
and colder climates, as the various productions of our 
earth require, and of such as afford a possibility of dis- 
covering it, we observe a regular motion round their 
axes, like that of our earth, causing an alternate reuirn 
of day and night, which is necessary for labour, rest, 
and vegetation, and that all parts of their surfaces may 
be exposed to the rays of the Sun. 

Such of the planets as are farthest from the Sun, and 
therefore enjoy least of his light, have that deficiency 
made up by several Moons, which constantly accompany 
and revolve about them, as our Moon revolves around the 
earth. The planet Saturn has over and above, a broad 
ring, encompassing it, which nowhere touches his body ; 
and which, like a broad zone in the Heavens, reflects the 
same light very copiously on that planet; if remote 
planets have the Sun's light fainter by day than we, they 
have an addition to it, morning and evening, by one or 
more of their Moons, and a greater quantity of light in 
the night time. 

On the surface of the Moon, (because it is nearer to us 
than any other of the celestial bodies,) we discover a 
nearer resemblance of our earth, for, by the assistance of 



12 OF ASTRONOMY IN GENERAL. [sEC. ti 

telescopes we observe the Moon to be full of high moun- 
tains, large valleys, and deep cavities. These similarities 
leave us no room to doubt, but that all planets, Moons, 
and Systems, are designed to be commodious habitations 
for creatures endowed with capacities of knowing, and 
adoring their beneficent Creator. 

Since the fixed stars are prodigious spheres shining by 
their own native light like our Sun, at inconceivable 
distances from each other, as well as from us, it is rea- 
sonable to conclude that they are made for similar pur- 
poses, each to bestow light, heat, and vegetation on a 
certain number of inhabited planets, kept by gravitation 
within the sphere of its activity. 

When we therefore contemplate on those ample and 
amazing structures, erected in endless magnificence over 
all the ethereal plains, when we look upon them as so 
many repositories of light, or fruitful abodes of life, when 
we consider that in all probability there are orbs vastly 
more remote than those which appear to our unaided 
sight, orbs whose effulgence, though travelling ever since 
the Creation, has not yet arrived upon our coast — 

What an august, what an amazing conception does 
this give of the works of the Omnipotent Creator ; who 
made use of no preparatory measures, or long circuit of 
means. He spake, and ten thousand times ten thousand 
Suns, multiplied without end, hanging pendulous in the 
great vault of Heaven, at immense distances from each 
other, attended by ten thousand times ten thousand 
worlds, all in rapid motion, yet calm, regular, and har- 
monious, invariably keeping the paths prescribed, rolled 
from his creating hand. 

But when we contemplate on the power, wisdom, 
goodness, and magnificence of the Great Creator ; let us 
use the language of the immortal Dr. Young, in his 
appeal to the starry Heavens : — 



" Say proud arch, 



Built with Divine ambition, iri disdain 
Of limit built ; built in the taste of Heaven, 
Vast concave, ample dome. Wast thou designed 
A meet apartment for the Deity 7 ? 



SEC. I.] OP ASTRONOMY IN GENERAL. 13 

Not so ; that thought alone thy state impairs, 
Thy lofty sinks, and shallows thy profound, 
And straightens thy diffusive." 

INTERROGATIONS FOR SECTION FIRST. 

What is Astronomy ? 

It is a mixed Mathematical Science, teaching the 
knowledge of the celestial bodies, their magnitudes, mo- 
tions, distances, periods, eclipses, and order. 

What are its uses ? 

What conviction does a knowledge of this branch of 
science give to the Understanding? 

What cheering sentiment is formed from a knowledge 
of this science ? 

What is the diameter of the earth ? 

How many miles is the diameter of the earth's orbit ? 

How is it known that the stars are at immense dis- 
tances from us ? 

How is it known that they are at immense distances 
from each other ? 

What instruments have been invented to aid the sight 
of man ? 

Who supposed there were stars, whose light had not 
yet reached the earth since their first creation ? 

Who confirmed the idea ? 

Why cannot the same rays be reflected back from the 
stars to our eyes ? 

With what light do the stars shine ? 

How could the Deity have given us greater light in 
the night time, than by the whole starry host ? 

How is it known that the Comets belong to the Solar 
System ? 

From what parity of reasoning, is it believed that the 
stars are so many suns, and have worlds revolving about 
them? 

How are those planets supplied with light, which are 
farthest from the sun ? 

2 



14 & THE SOLAR SYSTEM. [SEC. II. 



SECTION SECOND. 



OF THE SOLAR SYSTEM. 

The Solar system consists of the Sun, with all the 
Planets and Comets that move around him as their cen- 
tre. (See Frontispiece.) Those which are near the Sun, 
not only finish their circuits sooner, but likewise move 
with greater rapidity in their respective orbits, than 
those which are more remote. Their motions are all 
performed from West to East, in Elliptical orbits. Their 
names, distances, magnitudes, and periodical revolu- 
tions, are as follows :— The sun is placed near the com- 
mon centre, or rather in the lower focus of the orbits of 
all the planets and comets, and turns round on his axis 
once in 25 days, 14 hours, and 8 minutes ; as has been 
proved, from the motion of the spots, seen on his sur- 
face. His diameter is computed at 883,246 miles, and 
by the various attractions of the convolving planets, he 
is agitated by a small motion round the centre of gravity 
of the system. His mean apparent diameter as seen 
from the earth, is 32 minutes and one second. His so- 
lidity, and indeed that of every other planet, may be 
found by multiplying the cube of their diameters by 
,5236. All the planets as seen by a spectator, placed on 
the sun, move the same way, and according to the order 
of the signs, Aries, Taurus, Gemini, &c. which repre- 
sent the great ecliptic. But, to a spectator placed on 
any one of the planets, the others sometimes appear to 
go backward, sometimes forward, and at others sta- 
tonary ; not moving in proper circles, nor elliptical 
orbits, but in looped curves, which never return into 
themselves. 

Let S be the sun, (Plate 5th, fig. 1,) E the earth, M 



SEC. II.] OF THE SOLAR SYSTEM. 15 

Mercury, and V Venus. When the earth is at i, Mer- 
cury in his orbit at b, appears stationary at e. While 
Mercury is moving from b through his superior conjunc- 
tion at c to d, his motion appears direct among the fixed 
stars from e to f. At d his motion is imperceptible for a 
short time, when he appears stationary at f. As he 
passes from d through his inferior conjunction to b, his 
motion appears to be retrograde. At b he again appears 
stationary. 

The Comets, also appear to come from all parts, and 
appear to move in various directions. These proofs are 
sufficient to establish the fact, that the sun is placed near 
the centre, and that all the other planets revolve around 
him : are irradiated by his beams : receive the distribu- 
tion of his rays, and are dependant for the enjoyment 
of every blessing on this grand dispenser of divine 
munificence. 

The orbits of the planets are not in the same plane 
with the ecliptic,* but cross it in two points directly 
opposite to each other, called the planet's nodes.t — That 
from which the planet ascends northward above the 
ecliptic, is called the ascending node; and the other 
which is directly opposite, (and consequently 6 signs 
asunder,) is called the descending node. 

It was discovered on the first of January 1805, that 
the ascending node of the planet Herschel was in twelve 
degrees and fifty-three minutes of the sign Gemini, and 
advances 16 seconds in a year. Saturn in twenty-one 
degrees and 59 minutes of Cancer, and advances 32 se- 
conds in a year. Jupiter in 8 degrees and 27 minutes 
of Cancer, and advances 36 seconds yearly. Mars in 
18 degrees, and four minutes of Taurus, and advances 
28 seconds yearly. Yenus in 14 degrees and 55 minutes 
of Gemini, and advances 36 seconds yearly. Mercury 
in 16 degrees of Taurus, and advances 43 seconds every 

* The ecliptic is an imaginary great circle in the Heavens, in the 
plane of which the earth performs her annnal revolutions round the 
sun: 

t The node is the intersection of the orbit of any planet with that 
of the earth. 



16 OF THE SOLAR SYSTEM. [sEC. If. 

year. In these observations, the earth's orbit is con- 
sidered the standard, and the orbits of all the other pla- 
nets obliquely to it. The nearest planet to the sun is 
Mercury. The great brilliancy of light emitted by this 
planet : the shortness of the period during which obser- 
vations can be made upon his disk ; and his position 
among the vapours of the horizon when he is observed, 
have hitherto prevented Astronomers from making 
interesting discoveries to be relied on with certainty re- 
specting this planet. This planet, when viewed at dif- 
ferent times with a good telescope, appears in all the 
various shapes of the Moon, (See plate 3d, fig. 1, 2, 3,) 
which is a plain proof that he receives, (like the Moon,) 
all his light from the Sun. That he moves round the 
Sun in an orbit, within the orbit of the earth, is also 
plain ; because he is never seen opposite to the Sun, nor 
above 56 times the Sun's diameter from his centre. It 
has been said by authors, that his light and heat from 
the Sun must be almost seven times as great as ours ; 
judging from his nearness to it. His light and heat, 
however, depend more on the height and density of his 
atmosphere, than to his near approach to that luminary. 
His distance from the Sun is computed at 37,000,000 
of miles, is 3,225 in diameter, and performs a revolution 
round the Sun, in 87 days, 23 hours, 15 minutes and 28 
seconds : his apparent diameter as seen from the earth, 
is ten seconds. His orbit is inclined, 7 degrees to the 
ecliptic ; and that node from which he ascends north- 
ward above it, is in the 16th degree of Taurus ; the oppo- 
site in the 16th degree of Scorpio. The earth is in 
these points on the 6th of November, and the 4th of 
May ; when he comes to either of his nodes at his infe- 
rior conjunction about these times, he will appear to pass 
over the face of the sun like a dark round spot. — But in 
all other parts of his orbit, his conjunctions are invisible ; 
because he either goes above or below the Sun. On the 
5th day of May, at 6 hours, 43 minutes, 22 seconds in 
the morning, in the year 1832, in the longitude of 
Washington, he was in conjunction with the Sun. His 



SEC. II.] OF THE SOLAR SYSTEM. 17 

next visible conjunction will be on the 7th day of No- 
vember 1835. 

Venus, the next planet in order, is 68,000,000 of miles 
from the Sun by computation, and by moving at the 
rate of 69,000 miles every hour in her orbit, she per- 
forms her revolution round the Sun in 224 days, 16 
hours, and 49 minutes of our time. Her diameter is 
computed at 7687 miles, and performs her diurnal revo- 
lutions in 23 hours, 20 minutes and 54 seconds ; with 
an inclination of her orbit to the ecliptic, of 3 degrees, 
23 minutes, and 35 seconds. Her orbit includes the 
orbit of Mercury within it, for at her greatest elonga- 
tion, or apparent distance from the Sun, she is about 96 
times his diameter from his centre ; while that of Mer- 
cury is not above 56. 

Her orbit is included within the orbit of the earth, for 
if it were not, she would be as often seen in opposition 
as in conjunction with the sun. But she never departs 
from the Sun to exceed 47 degrees, and that of Mercury 
28, it is therefore certain that the orbit of Mercury is 
within the orbit of Venus, and that of Venus within the 
orbit of the earth. When this planet is west of the 
Sun, she rises in the morning before him, and hence 
she is called the morning star ; and when she sets after 
the Sun, she is called the evening star ; so that in one 
part of her orbit she rides foremost in the procession of 
night, and in the other, anticipates the dawn ; being 
each in its turn 290 days. The axis of Venus is inclined 
75 degrees to the axis of her orbit, which is 51 degrees 
and 32 minutes more than the axis of the earth is in- 
clined to the axis of the ecliptic ; and therefore her 
seasons vary much more than ours. The north pole 
of her axis, inclines towards the 20th degree of Aqua 
rius ; the earth's to the beginning of Cancer. Conse- 
quently the northern parts of Venus have Summer, in 
the signs where those of the earth have Winter, and 
vice versa. The orbit of Venus is inclined three and 
one half degrees to the earth's, and crosses it in the 14th 
degree of Gemini, and Sagitarius, and therefore when 
2*' 



18 OF THE SOLAR SYSTEM. [SEC. II. 

the earth is near the points of the ecliptic, at the time 
when Venus is in her inferior conjunction,* she appears 
like a spot on the Sun, and it furnishes a true method 
of calculating the distances of all the planets from the 

The surface of Venus, being enveloped in her atmos- 
phere, is probably the reason that so few spots have 
been seen on her disk. Plate 3, fig. 1st represents the 
spots on Venus discovered by Bianchini ; fig. 2d, those 
discovered by Dr. Herschel ; and figure 3d, represents the 
appearance with her blunt horn and rugged edge. 

It will not be uninteresting to those who peruse this 
treatise, to be put in the possession of all the elements of 
the transits, both of Mercury and Venus over the Sun's 
disk, from this period to the end of the present century. 
I therefore insert the following tables : — 

TRANSIT OF MERCURY OVER THE SUN'S DISK. 

Transit of Mercury, May 4th, 1832. 

D. h. m. s. 
Mean time of conjunction, May, . . . 4 23 51 22 

s. d. m. s. 
Geocentric longitude of the' Sun and Mercury, 1 14 56 45 

h. m. s. 

Middle apparent time, 18 1 

Semi-duration of the transit, . : . . 3 28 2 

Nearest approach to centres, ' 8 16 North. 

November 1th, 1835. 

H. M. S. 

Mean time of conjunction, . . . . 7 47 54 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 7 14 43 8 

h. m. s. 

Middle apparent time, 8 12 22 

Semi-duration of the transit, . . . . . 2 33 53 
Nearest approach of centres, . . . . 5 37 South. 

May Sth, 1845. ' 

H. M. S. 

Mean time of conjunction, . . . . . 7 54 18 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 1 18 1 49 

* Inferior conjunction is, when the planet is between the earth 
and the Sun in the nearest part of its orbit. 



SBC. II.] OF THE SOLAR SYSTEM. m 19 

H. M. 8. 

Middle apparent time, 7 32 58 

Semi-duration of the transit, . . . . 3 22 33 

Nearest approach of centres, • . . . 8 58 South. 

November 9th, 1848. 

H. M. S. 

Mean time of conjunction, 1 37 43 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 7 17 19 19 

h. m. s. 
Middle apparent time, . . . . . 1 49 43 

Semi-duration of the transit, . . . . 2 41 33 

Nearest approach of centres, .... 2 36 North. 

November Uth, 1861. 

H. M. S. 

Mean time of conjunction, . . . . . 19 20 13 

s. d. m. s: 
Geocentric longitude of the Sun and Mercury, 7 19 54 44 

h. m. s. 

Middle apparent time, 19 20 14 

Semi-duration of the transit, . . . 2 23 

Nearest approach of centres, . . . . 10 52 North. 

November <Wi, 1868. 

H. M. S. 

Mean time of conjunction.. . . . . . 18 43 45 

Middle apparent time, '. . ." . . 19 18 24 
Semi-duration of the transit, . . . . 1 45 21 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 7 13 9 42 
Nearest approach of centres, .... 12 20 South. 

May 6th, 1878. 

H. M. S. 

Mean time of conjunction, . . . . 6 38 30 

Middle apparent time, 6 55 14 

Semi-duration of the transit, 3 53 31 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 1 16 3 50 
Nearest approach of centres, .... 4 39 North. 

November 1th, 1881. 

H. M. S. 

Mean time of conjunction, . . . . . 12 39 38 

Middle apparent time, 12 59 33 

feemi-duration of the transit, . . . . . 2 39 6 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 7 15 46 57 
Nearest approach of centres, .... 3 57 South. 



20 OP THE SOLAR SYSTEM. [SEC II. 

May 9th, 1891. 

H. M. S. 

Meantime of conjunction, 14 44 57 

Middle apparent time, 14 13 46 

Semi-duration of the transit, . . . . 2 34 20 

S. D. M. S. 

Geocentric longitude of the Sun and Mercury, 1 19 9 1 
Nearest approach of centres, . . . . 12 21 South. 
November 10th, 1894. 

H. M. S. 

Meantime of conjunction, 6 17 5 

Middle apparent time, . . . . . G 36 29 

Semi-duration of the transit, . . . . 2 37 36 

s. d. m. s. 
Geocentric longitude of the Sun and Mercury, 7 18 22 9 
Nearest approach of centres. . . . . 4 20 North. 

TRANSITS OF VENUS OVER THE SUN'S DISK, FROM 
THE YEAR 1769 TO THE YEAR 2004 INCLUSIVE. 

June 3d, 1769. 

H. M. S. 

Mean time of conjunction, . . . 9 58 34 

Middle apparent time, . . . 10 27 3 

Duration of the transit, . . . 5 59 46 

s. d. m. s. 
Geocentric longitude of the Sun and Venus, . 2 13 27 8 
Nearest approach of centres, . . 10 10 North. 

December 8th, 1874. 

H. m. s. 
Mean time of conjunction, . . . 16 8 24 

Middle apparent time, . . . 15 43 28 

Duration of the transit, . . . 4 9 22 

s. d. m. s. 
Geocentric longitude of the Sun and Venus, . 8 16 57 49 
Nearest approach of centres, . . 13 51 North. 

December 6th, 1882. 

H. M. S. 

Mean time of conjunction, . . . 4 16 24 

Middle apparent time, . . . 4 49 42 

Duration of the transit, . . . 6 3 26 

s. d. m. s. 

Geocentric longitude of the Sun and Venus, . 8 14 29 14 

Nearest approach of centres, . 10 29 South. 
June 1th, 2004. 

H. M. S. 

Mean time of conjunction, . . . 20 51 24 

Middle apparent time, . . . 20 26 59 

Duration of the transit, . . . 5 29 40 

s. d. m. s. 
Geocentric longitude of the Sun and Venus, 2 17 54 23 

Nearest approach of centres, . . 11 19 South, 



SEC. II.] OF THE SOLAR SYSTEM. 21 

The Earth is the next planet above Venus, in the So- 
lar System ; it is 95,000,000 of miles from the Sun, and 
performs a revolution around him, from any Solstice, or 
Equinox, to the same again, in 365 days, 5 hours, and 
49 minutes : but from any fixed star to the same again, 
in 365 days, 6 hours, and 9 minutes. The former being 
the length of the tropical year, and the latter the sidereal. 
It travels at the rate of 58,000 miles every hour, in per- 
forming its annual revolution. It revolves on its own 
axis from West to East, once in 24 hours. Its mean 
diameter as seen from the Sun, is 17 seconds and two 
tenths of a degree ; which, by calculation, will give about 
7,970 miles for its diameter. The form of the Earth is 
an oblate spheroid, whose equatorial axis exceeds its po- 
lar by 36 miles, and is surrounded by an atmosphere 
extending 45 miles above its surface. 

The Seas, and unknown parts of the Earth, (by a 
measurement of the best Maps,) contain 160 millions, 
522 thousand, and 26 square miles. The inhabited parts 
38 millions, 990 thousand, 569. Europe four millions, 
456 thousand, and 65. Asia 10 millions, 568 thousand, 
823. Africa 9 millions, 654 thousand, 807. America 
14 millions, 110 thousand, 874 : the whole amounting 
to 199 millions, 512 thousand, 595; which is the num- 
ber of square miles on the whole surface of the Globe we 
inhabit. 

The Moon is not a planet, but only a satellite, or an 
attendant of the Earth, performing a revolution round it 
in 29 days, 12 hours, and 44 minutes ; and with the 
Earth, is carried round the Sun once in every year. 

The diameter of the Moon is 2.180 miles, and her 
mean distance from the Earth's centre, is estimated at 
240,000 miles. She goes round her orbit in 27 days, 7 
hours, and 43 minutes ; moving about 2,290 miles every 
hour, and performs a revolution on her own axis exactly 
in the time she performs a revolution in her orbit. — 
Let E be the Earth, ABCD the Moon's orbit, a a moun- 
tain on the side of the Moon, next the earth, (plate 5th> 
figure 2d.) As the Moon passes in her orbit from A to 



22 OF THE SOLAR SYSTEM. [SEC. II. 

B 90 degrees, it is evident she must turn on her axis 
ninety degrees, in order that the same side may be 
towards the Earth : the mountain will then be at b. — 
When the Moon is at C, having passed 180 degrees, half 
her revolution, the mountain must be at c. The Moon 
at D, presents the mountain at d. When the Moon re- 
turns to A, the mountain must come round to a again. 
Thus, in a sidereal revolution, the Moon must have re- 
volved once on its axis, or the same side could not have 
been presented to the earth ; consequently the same side 
of her is continually presented towards the Earth, and 
the length of her day and night taken together, is equal to 
same space of time. Her mean apparent diameter, as 
seen from the earth, is 31 minutes and 8 seconds of a de- 
gree. The orbit of the Moon, crosses the ecliptic in two 
opposite points, called the Moon's nodes, consequently 
one half of her orbit is above the ecliptic, and the other 
below ; the angle of its obliquity is 5 degrees and 20 
minutes. 

The Moon has scarcely any difference of seasons, be- 
cause her axis is nearly perpendicular to the ecliptic, and 
consequently the Sun never removes sensibly from her 
equator. 

The earth which we inhabit, serves as a satellite to 
the Moon, waxing and waning regularly, but appearing 
thirteen times as large, and affording her thirteen times 
as much light as the Moon does to us. When she is new 
to us, the earth appears full to her ; and, when she is in 
her first quarter as seen from the earth, the earth is in 
its third quarter as seen from the Moon. 

The Moon is an opaque globe, like the earth, and 
shines only by reflecting the light of the Sun ; therefore 
whilst that half of her which is towards the Sun, is en- 
lightened, the other half mus|; be dark and invisible. — 
Hence she disappears when she comes between us and 
the Sun ; because her dark side is then towards us. 

The planet Mars is next in order, being the first above 
the earth's orbit. His distance from the Sun is computed 
at 144,000,000 of miles ; and by travelling at the rate of 



SEC. II.] OP THE SOLAR SYSTEM. 23 

54,000 miles every hour, he goes round the Sun in 686 
of our days. 23 hours, and 30 minutes, which is the 
length of his year, equal to 667 and 3-4ths of his days ; 
and every day and night together, being nearly 40 
minutes longer than with us. His diameter is computed 
at 4,189 miles, and by his diurnal rotation, the inhabit- 
ants at his equator are carried 528 miles every hour. — 
The Sun appears to the inhabitants of Mars, nearly two- 
thirds the size that it does to us. 

His mean apparent diameter, as seen from the earth, 
is 27 seconds, and as seen from the Sun, ten seconds of 
of a degree. His axis is inclined to his orbit 59 degrees 
and 22 minutes. 

To the inhabitants of the planet Mars, our Earth and 
Moon appear like two Moons ; the one being 13 times 
as large as the other ; changing places with each other, 
and appearing sometimes horned, sometimes half or 
three-quarters illuminated, but never full, nor at most 
above one-quarter of a degree from each other ; although 
they are in fact 240,000 miles asunder. 

This Earth appears almost as large from Mars as 
Venus does to us. It is never seen above 48 degrees 
from the Sun, at that planet. Sometimes it appears to 
pass the disk of the Sun, and likewise Mercury and 
Venus. But Mercury can never be seen from Mars by 
such eyes as ours (unless assisted by proper instru- 
ments,) and Venus will be as seldom seen as we see 
Mercury. Jupiter and Saturn are as visible to Mars as 
to us. His axis is perpendicular to the ecliptic, and his 
inclination to it is one degree and 51 minutes. The 
planet Mars is remarkable for the redness of its light, the 
brightness of its polar regions, and the variety of spots 
which appear upon its surface. The atmosphere of this 
planet, which Astronomers have long considered to be 
of an extraordinary height and density, is the cause of 
the remarkable redness of its appearance. When a 
beam of white light passes through any medium, its 
colour inclines to redness, in proportion to the density of 
the medium ; and the space through which it has tra- 



•24 OF THE SOLAR SYSTEM. [SEC. II. 

veiled. The momentum of the red, or at least refran- 
gible rays being greater than that of the violet, or most 
refrangible, the former will make their way through the 
resisting medium, while the latter are either reflected or 
absorbed. The colour of the beam therefore when it 
reaches the eye, must partake of the colour of the least 
refrangible rays, and must consequently increase with 
the number of those of the violet, which have been ob- 
structed. — Hence we discover, that the morning and 
evening clouds are beautifully tinged with red, that the 
Sun, Moon and Stars appear of the same colour, when 
near the horizon, and that every luminous object seen 
through a mist, is of a ruddy hue. There is a great dif- 
ference of colour among the planets, we are therefore, (if 
the preceding observations be correct,) under the neces- 
sity of concluding, that those in which the red colour pre- 
dominates, are surrounded with the most extensive and 
dense atmospheres. According to this idea, the atmo- 
sphere of Saturn, must be the next to that of Mars, in 
density and extent. 

The planet Mars is an oblate spheroid, whose equa- 
torial diameter is to the polar as 1,355 is to 1,272, or 
nearly as 16 to 15. This remarkable flattening at the 
poles of Mars, probably arises from the great variation 
in the density of his different parts. To the inhabitants 
of the earth Mars appears sometimes gibbous, some- 
times full, but never horned. Figures 4, 5, 6 and 7, of 
plate third, represent different telescopic appearances of 
Mars. At figure 5 he appears gibbous. 

VESTA. 

Some Astronomers supposed that a planet existed be- 
tween the orbits of Jupiter and Mars ; judging from the 
regularity observed in the distances of the former dis- 
covered planets from the Sun. The discovery of Ceres 
confirmed this conjecture, but the opinion which it 
seemed to establish respecting the harmony of the Solar 
System, appeared to be completely overturned, by the 
discovery of Pallas and Juno. Dr. Olbers, however, 



SEC. II.] OF THE SOLAR SYSTEM. 25 

imagined that these small celestial bodies were merely 
the fragments of a larger planet which had burst asun- 
der by some internal convulsion, and that several more 
might yet be discovered between the orbits of Mars and 
Jupiter. He therefore concluded that though the orbits 
of all these fragments might be inclined to the ecliptic, 
yet as they must have all diverged from the same point, 
they ought to have two common points of reunion, or 
two nodes in opposite regions of the Heavens, through 
which all the planetary fragments must sooner or later 
pass. One of these nodes he found to be in Virgo, and 
the other in the Whale, and it was actually in the latter 
of these regions, that Mr. Harding discovered the planet 
Juno. With the intention therefore of detecting other 
fragments of the supposed planet, Dr. Olbers examined 
thrice every year all the little stars in the opposite con- 
stellations of the Virgin and the Whale, till his labours 
were crowned with success on the 29th of March, 1807, 
by the discovery of a new planet in the constellation 
Virgo, to which he gave the appropriate name of Vesta. 
The planet Vesta is the next above Mars, and is in ap- 
pearance of the fifth or sixth magnitude, and may be 
seen in a clear morning by the naked eye. Its light is 
more intense, pure and white, than either of the three 
following, Ceres, Juno, or Pallas. Its distance from the 
Sun is computed at 225 millions of miles, and its diame- 
ter at 238 : its revolutions have not hitherto been suffi- 
ciently ascertained. 

ON JUNO. 

The planet Juno, the next above Vesta, and between 
the orbits of Mars and Jupiter, was discovered by Dr. 
Harding, at the Observatory near Bremen, on the even- 
ing of the 5th of September, 1804. This planet is of a 
reddish colour, and is free from that nebulosity which 
surrounds Pallas. It is distinguished from all the other 
planets by the great eccentricity of its orbit, and the 
effect of this is so extremely sensible, that it passes over 
that half of its orbit which is bisected by its perihelion in 

3 



26 OF THE SOLAR SYSTEM. [SEC II. 

half the time that it employs in describing the other half, 
which is further from the Sun : from the same cause its 
greatest distance from the Sun is double the least. The 
difference between the two being about 127 millions of 
miles. Its mean distance from the Sun is computed at 
252 millions of miles, and performs its tropical revolu- 
tion in 4 years and 128 days. Its diameter is estimated 
at 1,425 miles, and its apparent diameter as seen from 
the Earth, three seconds of a degree, and its inclination 
of orbit twenty-one degrees, 

ON CERES. 

The planet Ceres was discovered at Palermo, in Si- 
cily, on the first of January, 1801, by M. Piazzi. an inge- 
nious observer, who has since distinguished himself by 
his Astronomical labours. It was however again dis- 
covered by Dr. Olbers, on the first of January, 1807, 
nearly in the same place where it was expected from the 
calculations of Baron Zach. The planet Ceres is of a 
ruddy colour, and appears about the size of a star of the 
8th magnitude. It seems to be surrounded with a large 
dense atmosphere of 675 miles high, according to the 
calculations of Schroeter, and plainly exhibits a disk, 
when examined with a magnifying power of 200. 

Ceres is situated between the orbits of Mars and Jupi- 
ter, She performs her revolution round the Sun in four 
years, 7 months and ten days ; and her mean distance 
is estimated at 263 millions of miles from that luminary. 
The observations which have been hitherto made upon 
this celestial body, do not appear sufficiently correct to 
determine its magnitude with any degree of accuracy. 

ON PALLAS. 

The planet Pallas was discovered at Bremen, in Lower 
Saxony, on the evening of the 28th of March, 1802, by 
Doctor Olbers, the same active Astronomer who re-dis- 
covered Ceres. It is situated between the orbits of Mars 
and Jupiter, and is nearly of the same magnitude and 
distance with Ceres, but of a less ruddy colour. It is 



SEC. II.] OF THE SOLAR SYSTEM. 27 

seen surrounded with a nebulosity of almost the same 
extent, and performs its annual revolution in nearly the 
same period. The planet Pallas, however, is distin- 
guished in a very remarkable manner from Ceres and 
all the primary planets, by the immense inclination of 
its orbit. While these bodies are revolving round the 
Sun in almost circular paths, rising only a few degrees 
above the plane of the ecliptic, Pallas ascends above this 
plane, at an angle of about 35 degrees, which is nearly" 
five times greater than the inclination of Mercury. From 
the eccentricity of Pallas being greater than that of Ceres, 
or from a difference of position in the line of their apsides, 
where their mean distances are nearly equal, the orbits 
of these two planets mutually intersect each other : [see 
Frontispiece :] a phenomenon which is altogether anoma- 
lous in the Solar System. 

Pallas performs its tropical revolution in four years, 7 
months, and 11 days. The distance of this planet from 
the Sun, is estimated at 265 millions of miles. It is sur- 
rounded with an atmosphere 468 miles high. 

OF JUPITER. 

Jupiter, the largest of all the planets, is still higher in 
the Solar System, being four hundred and ninety millions 
of miles from the Sun, and by performing his annual 
revolution round the Sun in 11 years, 314 days, 20 hours, 
and 27 minutes, he moves in his orbit at the rate of 29,000 
miles in an hour. The diameter of this planet is esti- 
mated at 89,170 miles, and performs a revolution on its 
own axis in nine hours, 55 minutes, and 37 seconds ; 
which is more than 28,000 miles every hour, at his equa- 
tor, the velocity of motion on his axis being nearly equal 
to the velocity with which he moves in his annual 
orbit. 

This planet is surrounded by faint substances called 
belts, in which so many changes appear, that they have 
been regarded by some, as clouds or openings in the at- 
mosphere of the planet ; while others imagine that they 
are of a more permanent nature, and are the marks of 



28 OF THE SOLAR SYSTEM. [SEC. II, 

great physical revolutions which are perpetually chang- 
ing the surface of the planet. 

In 1700, May 28th, the disk of Jupiter was observed 
by Dr. Herschel, covered with small curved belts, or 
rather lines, not contiguous, as in plate 3d, figures 8 and 
9. Parallel belts, however, as represented in plate third, 
fig. 10, are most common. 

The axis of Jupiter is so nearly perpendicular to his 
orbit, that he has no sensible change of seasons, which is 
a great advantage, and wisely ordered by the Author of 
nature ; for if the axis of this planet were inclined any 
considerable number of degrees, just so many degrees 
round each pole would in their turn, be almost six of 
our years together in darkness, and, as each degree of a 
great circle on Jupiter contains 778 of our miles at a 
mean rate : judge ye what vast tracts of land would be 
rendered uninhabitable by any considerable inclination 
of his axis. 

The difference between the equatorial and polar di- 
ameters of this oblate spheroid is computed at 6,230 
miles ; for his equatorial diameter is to his polar, as 13 
is to 12 ; consequently his poles are 3,115 miles nearer 
his centre than his equator. This results from his rapid 
motion round his axis, for the fluids together with the 
light particles which they can carry, or wash away with 
them, recede from the poles, which are at rest towards 
the equator, where the motion is more rapid, until there 
be a sufficient number of such particles accumulated to 
make up the deficiency of gravity occasioned by the cen- 
trifugal force, which arises from a quick motion round 
an axis ; and when the deficiency of weight or gravity 
of the particles is made up by a sufficient accumulation, 
the equilibrium is restored, and the equatorial parts rise 
no higher. The orbit of Jupiter is inclined to the eclip- 
tic one degree and twenty minutes. His north node is 
in the 7th degree of Cancer, and his south node in the 
7th degree of Capricorn. His mean apparent diameter 
as seen from the earth is 39 seconds, and as seen from 
the Sun, 37 seconds of a degree. 



SEC. II.J OF THE SOLAR SYSTEM. 29 

This planet being situated at so great a distance from 
the Sun, does not enjoy that degree of light emanating 
from his rays, which is enjoyed by the earth. To sup- 
ply this deficiency, the great Author of our existence has 
provided four satellites, or moons, to be his constant 
attendants, which revolve around him, in such manner, 
that scarcely any part of this large planet but is enlight- 
ened during the whole night, by one or more of these 
Moons, except at his poles, where only the farthest Moons 
can be seen ; there, however, this light is not wanted ; 
because the Sun constantly circulates in or near the hori- 
zon, and is very probably kept in view of both poles by 
the refraction of his atmosphere. The first Moon, or 
that nearest to Jupiter, performs a revolution around him 
in one day, 18 hours, and 36 minutes, of our time, and is 
229 thousand miles distant from his centre : the second 
performs his revolution in three days, 13 hours, and 15 
minutes, at a distance of 364 thousand miles : the third 
in seven days, 3 hours, and 59 minutes, at the distance of 
580 thousand miles : and the fourth, or farthest from his 
centre, in 16 days, 18 hours, aud 30 minutes, at the dis- 
tance of one million of miles from his centre. The angles 
under which these satellites are seen from the earth, as 
its mean distance from Jupiter, are as follow : — The first 
three minutes and 55 seconds : the second six minutes 
and 15 seconds : the third 9 minutes and 58 seconds, and 
the fourth 17 minutes and 30 seconds. This planet when 
seen from its nearest Moon, must appear more than one 
thousand times as large as our Moon does to us. 

The three nearest Moons to Jupiter, pass through his 
shadow, and are eclipsed by him, in every revolution, but 
the orbit of the fourth is so much inclined, that it passes 
by its opposition to Jupiter without entering his shadow, 
two years in every six. By these eclipses, Astronomers 
have not only discovered that the Sun's light is about 8 
minutes in coming to us ; but they have also determined 
the longitude of places on this earth with greater cer- 
tainty and facility, than bv any other method yet known. 

3* 



30 OF THE SOLAR SYSTEM. [SEC II. 

OF SATURN. 

Saturn is the most remarkable of all the planets ; it 
is calculated at 9 hundred millions of miles from the Sun, 
and travelling at the rate of 21,900 miles every hour, 
and performs' its annual circuit in 29 years, 167 days, 
and 2 hours of our time : which makes only one year 
to that planet. Its diameter is computed at 79,042 miles, 
and performs a revolution on its own axis once in ten 
hours, 16 minutes and two seconds. Its mean apparent 
diameter as seen from the earth, is 18 seconds, and as 
seen from the Sun, 16 seconds of a degree ; its axis is 
supposed to be 60 degrees inclined to its orbit. 

This planet is surrounded by a thin broad ring, which 
nowhere touches its body, and when viewed by the aid 
of a good telescope appears double. In Plate 2d, Sa- 
turn and his double ring are represented as in the largest 
view when seen from the earth. In Plate 3d, fig. 11, 
he appears as if viewed by a spectator at right angles 
to the plane of the ring. In Plate 3d, fig. 12, the ring 
is represented very obliquely to the view of the observer. 
It is inclined 30 degrees to the ecliptic, and is about 21 
thousand miles in breadth ; which is equal to its dis- 
tance from Saturn on all sides. This ring performs a 
revolution on its axis in the same space of time with the 
planet, namely, ten hours, 16 minutes and two seconds. 
— This ring seen from the planet Saturn, appears like a 
vast luminous circle in the Heavens, and, as if it does 
not belong to the planet. When we see the ring most 
open, its shadow upon the planet is broadest, and from 
that time the shadow grows narrower, as the ring ap- 
pears to do to us, until by Saturn's annual motion, the 
Sun comes to the plane of the ring, or even with its 
edge ; which being then directed towards the earth, it 
becomes to us invisible on account of its thinness. — 
The ring nearly disappears twice in every annual revo- 
lution of Saturn, when he is in the 19th degree, both 
of Pisces and Virgo. But, when Saturn is in the 19th 
degree either of Gemini or Sagitarius, his ring appears 



SEC. II.] OF THE SOLAR SYSTEM. 31 

most open to us, and then its longest diameter is to its 
shortest as 9 to 4. 

This planet is surrounded with no less than seven 
satellites, which supply him with light during the ab- 
sence of the Sun. The fourth of these was first dis- 
covered by Huygens, on the 25th of March, 1655. — 
Cassini discovered the fifth in October, 1671. The 
third on the 23d of December, 1672 : And the first and 
second in the month of "March, 1684. The sixth and 
seventh were discovered by Dr. Herschel in the year 
1789. These are nearer to Saturn than any of the 
others. 

These Moons perform their revolutions round this 
planet on the outside of his ring, and nearly in the 
same plane with it. The first, or nearest Moon to Sa- 
turn, performs its periodical revolution around him in 
22 hours and 37 minutes, at the distance of 121,000 
miles from his centre ; the second performs its periodi- 
cal revolution in one day, 8 hours, and 53 minutes, at 
the distance of 156 thousand miles ; the third in one 
day, 21 hours. 18 minutes, and 26 seconds, at the dis- 
tance of 193 thousand ; the fourth in two days, 17 hours, 
44 minutes, and 51 seconds, at the distance of 247 thou- 
sand ; the fifth in 4 days, 12 hours, 25 minutes, and 
11 seconds, at 346 thousand ; the 6th in 15 days, 22 
hours, 41 minutes and 13 seconds, at the distance of 802 
thousand ; and the 7th or outermost in 49 days, 7 hours, 
53 minutes, and 43 seconds, at the distance of two mil- 
lions, 337 thousand miles from the centre of Saturn — 
their primary. 

When we look with a good telescope, at the body of 
Saturn, he appears like most of the other planets, in the 
form of an oblate spheroid, arising from the rapid rota- 
tion about his axis. He however appears more flattened 
at the poles, than any of the others, and although his 
motion on his axis is not equal to that of Jupiter, yet he 
does not appear to be in form, so near that of a globe as 
that planet. When we consider that the ring by which 
Saturn is encompassed, lies in the same plane of his 



32 OF THE SOLAR SYSTEM. [SEC II. 

equator, and, that it is at least equal if not more dense 
than the planet, we shall find no difficulty in accounting 
for the great accumulation of matter, at the equator of 
Saturn. The ring acts more powerfully upon the equa- 
torial regions of Saturn, than upon any part of his disk \ 
and by diminishing the gravity of these parts, it aids the 
centrifugal force in flattening the poles of the planet. 
Had Saturn indeed never revolved upon his axis, the 
action of the ring would of itself have been sufficient, 
to have given it the form of a spheroid. 

The following, are the dimensions of this luminous 
zone, as determined by Dr. Herschel: miles. 

Inside diameter of the interior ring, . . 146,345 
Outside diameter of the interior ring, . 184,383 

Inside diameter of the exterior ring, . . 190,240 
Outside diameter of the exterior ring, . 204,833 

Breadth of the interior ring, . . 20,000 

Breadth of the exterior ring, . . . . 7,200 
Breadth of the dark space between the two rings, 2,839 
Angle which it subtends when seen at the mean m. s. 

distance of the planet. . • . . . 7 25 

ON HERSCHEL, OR URANUS. 

From inequalities in the motion of Jupiter and Saturn 
for which no rational account could be given, and from 
the mutual action of these planets, it was inferred by 
some Astronomers, that another planet existed beyond 
the orbits of Jupiter and Saturn ; by whose action these 
irregularities were produced. This conjecture was 
confirmed on the 13th of March, 1781 ; when Dr. Her- 
schel discovered a new planet, which in compliment to 
his Royal Patron, he called Georgium Sidus, although 
it is more generally known by the name of Herschel, or 
Uranus. This new planet, (which had been formerly 
observed as a small star by Flamsted, and likewise by 
Tobias Mayer, and introduced into their catalogue of 
fixed stars,) is situated, one thousand, eight hundred 
millions of miles from the centre of the System, and 
performs its revolution round the Sun in 83 years, 150 



SEC. II.] OF THE SOLAR SYSTEM. 33 

days, and 18 hours. Its diameter is computed at 
35,112 miles. When seen from the earth, its mean ap- 
parent diameter is three and I seconds, and as seen from 
the Sun, 4 seconds of a degree. As the distance of this 
planet from the Sun is twice as great as that of Saturn, 
it can scarcely be distinguished without the aid of in- 
struments. When the sky however is serene, it appears 
like a fixed star of the sixth magnitude, with a bluish 
white light, and a brilliancy between that of Venus and 
the Moon ; but seen with a power of two or three hun- 
dred, its disk is visible and well defined. — The want of 
light arising from the distance of this planet from the 
Sun is supplied by six satellites, all of which were dis- 
covered by Dr. Herschel. 

The first of those satellites is twenty-five and a half 
seconds from its primary, and revolves round it in 5 
days, 21 hours, and 25 minutes ; the second is nearly 
34 seconds distant from the planet, and performs its re- 
volution in 8 days, 17 hours, 1 minute and 19 seconds. 
The distance of the third satellite is 38,57 seconds, and 
the time of its periodical revolution is ten days, -23 
hours, and four minutes. The distance of the fourth 
satellite is 44,22 seconds, and the time of its periodical 
revolution is 13 days, 11 hours, 5 minutes and 30 se- 
conds. The distance of the fifth is one minute and 28 
seconds, and its revolution is completed in 38 days, 1 
hour, and 49 minutes. The sixth satellite, or the fur- 
thest from the centre of its primary, is at the distance of 
two minutes, and nearly 57 seconds, and therefore re- 
quires 107 days, 16 hours, and 40 minutes to complete 
one revolution. The second and fourth of these were 
discovered on the 11th of January, 1787, the other four 
were discovered in 1790, and 1794 ; but their distances, 
and times of periodical revolution, have not been so ac- 
curately ascertained as the other two. It is, however, a 
remarkable circumstance, that the whole of these satel- 
lites move in a retrograde direction, and in orbits lying in 
the same plane, and almost perpendicular to the ecliptic. 

When the Earth is in its perihelion, and Herschel in 



34 OF THE SOLAR SYSTEM. [SEC. IT. 

its aphelion, the latter becomes stationary, as seen from 
the Earth, when its elongation, or distance from the Sun 
is 8 signs, 17 degrees, and 37 minutes, his retrograda 
tions continue 151 days, and 12 hours. When the 
Earth is in its aphelion, and Herschel in its perihelion, 
it becomes stationary, at an elongation of 8 signs, 16 de- 
grees, and 27 minutes, and its retrogradations continue 
149 days and 18 hours. 

ON COMETS. 

Comets are a class of celestial bodies, which occa- 
sionally appear in the Heavens. They exhibit no visi- 
ble or defined disk, but shine with a pale and cloudy 
light, accompanied with a tail or train, turned from the 
Sun. They traverse every part of the Heavens, and 
move in every possible direction. 

When examined through a good telescope, a Comet 
resembles a mass of aqueous vapours, encircling an 
opaque nucleus, of different degrees of darkness in dif- 
ferent Comets ; though sometimes, as in the case of 
several discovered by Dr. Herschel, no nucleus can be 
seen. 

As the Comet advances towards the Sun, its faint and 
nebulous light becomes more brilliant, and its luminous 
train gradually increases in length. 

When it reaches its perihelion, the intensity of its 
light, and the length of its tail reach their maximum, 
and then it sometimes shines with all the splendour of 
Venus. During its retreat from the perihelion, it is 
shorn of its splendour, and it gradually resumes its nebu- 
lous appearance ; and its train decreases in magnitude, 
until it reaches such a distance from the Earth, that the 
attenuated light of the Sun which it reflects, ceases to 
make an impression on the organ of sight. Traversing 
unseen the remote portion of its orbit, the Comet wheels 
its etherial course far beyond the limits of the Solar 
System. What region it there visits, or upon what 
destination it is sent, the limited powers of man are un- 
able to discover. After the lapse of years, we perceive 



SEC. II.] OF THE SOLAR SYSTEM. 35 

it again returning to our System, and tracing a portion 
of the same orbit round the Sun, which it had formerly 
described. 

Various opinions have been entertained by Astrono- 
mers respecting the tails of Comets. These tails or 
trains, sometimes occupy an immense space in the Hea- 
vens. The Comet of 1681, stretched its tail across an 
arch of 104 degrees ; and the tail of the Comet of 1769 
subtended an angle of 60 degrees at Paris, 70 at Bo- 
logna, 97 at the Isle of Bourbon, and 90 degrees at Sea, 
between Teneriffe and Cadiz. These long trains of 
light are maintained by Newton, to be a thin vapour, 
raised by the heat of the Sun from the Comet. 

If we knew their uses in our System, we could form 
more probable conjectures as to the chronology of their 
creation. They have been noticed from the earliest era 
of our Astronomical History, and if our modern Philo- 
sophers had not discovered, that some (at least.) leave 
us to return again into our System, and therefore de- 
scribe a vast elliptical orbit round our Sun, we might 
have fancied that the periods of their first recorded ap- 
pearances in our field of science, were the eras of their 
individual formation. But their recurring presence 
proves, that their first existence ascends into unexplored 
and unrecorded antiquity. Yet, from whence they 
came to us, we as little know as for what purpose. The 
Comet of 16S2, re-appeared in 1759, in the interval de- 
scribing an orbit in the form of an ellipsis, answering to 
a revolution of 27,937 days. It will therefore re-appear 
in November, 1835. In its greatest distance, it is sup- 
posed not to go above twice as far as Uranus. This is 
indeed a prodigious sweep of space, and it has been 
justly observed, that the vast distance to which some 
Comets roam, proves how very far the attraction of the 
Sun extends; for though they stretch themselves to 
such depths in the abyss of space, yet by Virtue of the 
Solar power, they return into its effulgence. But it has 
been recently discovered, that three Comets (at least,) 
never leave the planetary system. One whose period is 



36 OP THE SOLAR SYSTEM. [SEC It. 

three years and a quarter, is included within the orbit 
of Jupiter ; another of six years and three quarters, ex- 
tends not so far as Saturn ; and a third of twenty years, 
is found not to pass beyond the circuit of Uranus. 

The transient effect of a Comet passing near the 
Earth, could scarcely amount to any great convulsion, 
but if the Earth were actually to receive a shock by 
collision, from one of those bodies, the consequences 
would be awful. A new direction would be given to its 
rotatory motion, and the Globe would revolve around a 
new axis. The Seas, forsaking their ancient beds, 
would be hurled by their centrifugal force to the new 
equatorial regions, islands and continents : the abodes of 
men and animals, would be covered by the universal 
rush of the waters to the new Equator, and every 
vestige of human industry and genius at once destroyed. 

Although the orbits of all the planets in the Solar 
System be crossed by five hundred different Comets, the 
chances against such an event however, are so very 
numerous, that there need be no dread of its occurrence ; 
besides, that Almighty Arm which first created them, 
and described for them their various orbits — that Om- 
nipotent Wisdom which directed the times of their pe- 
riodical revolutions, still continues to guide and protect 
all the workmanship of his hands. 

INTERROGATIONS FOR SECTION SECOND. 

Of what does the Solar System consist ? 

What planets finish their circuits soonest ? 

Which moves with the greatest rapidity ? 

In what direction do they move in their orbits ? 

What is the form of the orbits described by the 
planets 7 

Where is the Sun placed 1 

In what time does he turn round on his own axis ? 

How is that proved 1 

What is his diameter ? 

What is his mean apparent diameter as seen from the 
Earth ? 



SEC. II.] OF THE SOLAR SYSTEM. 37 

How is his solidity calculated ? 

What is the Ecliptic ? 

What is meant by the Nodes ? 

Which is the Ascending Node 7 

Which is the Descending Node 7 

How many signs are they asunder \ 

What additional Astronomical discoveries were made 
in the year 1805 ? 

What planet is nearest the Sun 7 

What reasons are given to suppose that this planet re- 
ceives its light from the Sun ? 

What is the computed distance of Mercury from the Sun 7 

What is its diameter 1 

In what time does it perform a revolution around the 
Sun? 

What time on its own axis ? 

How many miles does it move in an hour in its mo- 
tion round the Sun 7 

How many degrees is his orbit inclined towards the 
Ecliptic ? 

What planet is next to Mercury 7 

What is the distance of this pla.net from the Sun 7 

How many miles in an hour, does Venus move in 
performing her revolution round the Sun 7 

In what time does she perform her annual revolution 7 

In what time does she perform a revolution on her axis 7 

What is her diameter 7 

How is it known that the orbits of Mercury and Ve- 
nus are included within that of the Earth ? 

How many degrees at most does Mercury depart from 
the Sun 7 

How many Venus 7 

What is meant by inferior conjunction 7 

What is a transit 7 

What is the name of the third planet from the Sun 7 

What is its distance from the Sun 7 

What is its diameter 7 

In what time does it perform a revolution around the 
Sun? 

4 



38 OF THE SOLAR SYSTEM. [SEC II. 

What is its hourly progress ? 

In what time does it perform a revolution on its axis ? 

What is its form ? 

How many miles difference in the two diameters ? 

What is the Moon 7 

In what time does she perform a revolution round the 
Sun? 

Around the Earth ? 

Around her own axis ? 

What is her distance from the Earth ? 

How many miles in diameter ? 

What is her mean apparent diameter as seen from the 
Earth ? 

Do the orbits of the Earth and Moon coincide ? 

How much more light does the Earth give to the 
Moon, than the Moon gives to us ? 

What is the name of the fourth planet from the Sun ? 

What is his distance from that luminary ? 

In what time does he perform his annual reyolution ? 

What is his hourly progress 7 

What time his revolution on his axis 7 

What is his mean apparent diameter as seen from the 
Earth? 

What as seen from the Sun ? 

For what is the planet Mars remarkable ? 

What have Astronomers concluded to be the cause of 
this remarkable appearance ? 

What is its form ? 

W T hat is the name of the fifth planet from the Sun? 

By whom was it discovered ? And when ? 

In what sign of the Ecliptic can this planet be seen 
without the aid of a telescope ? 

What is its distance from the Sun ? 

What is its diameter ? 

What is the name of the sixth ? 

By whom was it discovered ? And when ? 

What is its colour? 

For what is it distinguished? 

What is its distance from the Sun ? 



SEC. II.] OF THE SOLAR SYSTEM. 39 

What is the time of its tropical revolution ? 

What is its diameter? 

What is 'its apparent diameter as seen from the Sun? 

What is the name of the seventh ? 

By whom was it first discovered ? 

In what year ? 

What is the height of its atmosphere? 

In what time does this planet perform its revolution 
round the Sun ? 

What is her distance from that luminary? 

What is the name of the eighth planet from the Sun ? 

When was it discovered ? And by whom ? 

What is its distance from the Sun ? 

In what time does it perform its annual revolution 
around him ? 

What is the height of its atmosphere ? 

By what name are the last four collectively called ? 

They are called Asteroids. 

What is the name of the ninth planet from the Sun ? 

How many miles distant from the Sun ? 

What is his diameter ? 

In what time does he perform his annual revolution 1 

In what time on his own axis ? 

What is his mean motion in his orbit ? 

What is his mean motion on his axis ? 

Have the inhabitants of Jupiter any sensible change 
of seasons ? 

How many miles constitute a degree on this planet ? 

How many miles difference between his equatorial 
and polar diameters ? 

Why is this great difference ? 

How many degrees is the orbit of Jupiter inclined to 
the Ecliptic? In what sign of the Zodiac is his north 
node ? In what sign his south node ? 

How many satellites attend this planet ? What is his 
apparent diameter as seen from the Sun ? What as seen 
from the Earth ? 

Of what benefit have those Moons been to the inhabit- 
ants of this Earth ? Can either of them be seen by us 
without the aid of telescopes ? 



40 OF THE SOLAR SYSTEM. [SEC. II. 

What name is given to the next, or tenth planet from 
the Sun? 

What is its distance from the Sun ? 

How many miles does this planet move in an hour ? 

In what time does it perform its revolution around the 
Sun ? In what time on its own axis ? 

What is its diameter ? 

What is its mean apparent diameter as seen from the 
Sun ? What as seen from the Earth ? 

How many degrees is its axis inclined to its orbit ? 

What encircles his body ? 

How does it appear when viewed with a telescope ? 

What is its breadth ? 

How does it appear to the inhabitants of Saturn ? 

Why is it sometimes invisible to us ? 

How many times does it appear in one revolution of 
the planet ? 

In what signs and degrees of the Ecliptic does it dis- 
appear ? 

In what signs and degrees does this ring appear most 
open? 

How many Satellites has this planet 1 

Where are they situated, inside or outside of the ring ? 

What is the form of this planet ? 

What is the name of the next, or outermost planet ? 

When was it discovered ? 

How far is it situated from the Sun ? 

In what time does it perform its annual revolution ? 

What is its diameter ? 

What is its apparent diameter as seen from the Sun ? 
What as seen from the Earth ? 

Can it be seen without the aid of a telescope ? 

How many Satellites attend it ? 

In what direction do those Satellites move ? 

What are Comets ? 

In what direction do they move ? 

In what part of its orbit is its train most brilliant ? 

What was Newton's opinion concerning the Comet's; 
tail, or train ? 



SEC. II.] OP THE SOLAR SYSTEM. 41 

How many Comets are supposed to belong to the So- 
lar System ? 

How many are known not to exceed the circuit of 
Uranus ? 

What would be the result if a Comet should come in 
actual collision with this Earth ? 

Is it probable that there will ever be such an occur- 
rence ? 

Why is it not probable 7 



42 ON GRAVITY. fsEC. 



III. 



SECTION THIRD 



ON GRAVITY. 



The power by which bodies fall towards the Earth, 
is called Gravity, or Attraction. By this power in the 
Earth it is, that all the bodies on whatever side, fall in 
lines perpendicular to its surface. On opposite parts of 
the Earth, bodies fall in opposite directions, all towards 
the centre, where the whole force of gravity appears to 
be accumulated. By this power constantly acting on 
bodies near the Earth, they are kept from leaving it, and 
those on its surface are kept by it, that they cannot fall 
from it. Bodies thrown with any obliquity, are drawn 
by this power from a straight line into a curve, until they 
fall to the ground. The greater the force with which 
they are projected, the greater is the distance they are 
carried before they fall. If we suppose a body carried 
several miles above the surface of the Earth, and there 
projected in a horizontal direction, with so great a velo- 
city that it moves more than a semidiameter of the 
Earth in the line it would take to fall to the Earth by 
gravity, in that case, if there were no resisting medium, 
the body would not fall to the Earth at all ; but continue 
to circulate round the Earth, keeping always the same 
path, and returning to the point from whence it was pro- 
jected with the same velocity with which it moved at 
first. We find that the Moon therefore must be acted 
upon by two powers, one of which would cause her to 
move in a right line, another bending her motion from 
that line into a curve. This attractive power must be 
seated^ in the Earth, for there is no other body within the 



SEC. 



III.l ON GRAVITY. 43 



Moon's orbit to draw her.* The attractive power of the 
Earth therefore extends to the Moon, and in combination 
with her projectile force, causes her to move round the 
Earth in the same manner as the circulating body above 
supposed. 

The Moons of Jupiter, Saturn and Herschel, are ob- 
served to move around their primary planets ; therefore 
there is an attractive power in these planets, operating 
on their Satellites in the same manner as the attraction 
of the Earth operates on the Moon. All the planets and 
Comets move round the Sun, and respect it as their 
centre of motion, therefore the Sun must be endowed 
with an attracting power, as well as the Earth and pla- 
nets. Consequently all the bodies, or matter of the Solar 
System are possessed of this attractive power, and also 
all matter whatsoever. 

As the Sun attracts the planets with their Satellites, 
and the Earth the Moon, so the planets and Satellites 
re-attract the Sun, and the Moon the Earth. This is 
also confirmed by observation ; for the Moon raises 
tides in the ocean ; the satellites and planets disturb 
each other's motions. Every particle of matter being 
possessed of an attractive power, the effect of the whole 
must be in proportion to the quantity of matter in the 
body. 

Gravity also, like all other virtues, or emanations, 
either drawing or impelling a body towards a centre, de- 
creases as the square of the distance increases ; that is, 
a body at twice the distance, attracts another with only 
a fourth part of the force ; at four times the distance, 
with a sixteenth part of the force. 

By considering the law of gravitation which takes 
place throughout the Solar System, it will be evident 

* If the Moon revolves in her orbit in consequence of an attractive 
power residing in the Earth, she ought to be attracted as much from 
the tangent of her orbit in a minute, as heavy bodies fall at the 
Earth's surface in a second of time. It is accordingly found by cal- 
culation, that the Moon is deflected from the tangent 16,09 feet in a 
minute, which is the very space through which heavy bodies descend 
in a second of time at the Earth's surface. 



44 ON GRAVITY. [SEC. HI. 

that the Earth moves round the Sun in a year. It has 
been stated and shown, that the power of gravity de- 
creases as the square of the distance increases, and from 
this it follows with mathematical certainty, that when 
two or more bodies move round another as their centre 
of motion, the squares of the time of their periodical re- 
volutions, will be in proportion to each other, as the 
cubes of their distances from the central body. — This 
holds precisely with regard to the planets round the Sun, 
and the satellites round their primaries, the relative dis- 
tances of which are well known. 

All globes which turn on their own axis, will be ob- 
late spheroids, that is, their surfaces will be further from 
their centres in the equatorial than in the polar regions ; 
for as the equatorial parts move with greater velocity, 
they will recede farthest from the axis of motion, and 
enlarge the equatorial diameter. That our Earth is 
really of this figure, is demonstrable from the unequal 
vibrations of a pendulum, and the unequal length of de- 
grees in different latitudes. 

Since then the Earth is higher at the equator than at 
the poles, the seas naturally would run towards the 
polar regions, and leave the equatorial parts dry, if the 
centrifugal force of these parts, by which the waters 
were carried thither, did not keep them from returning. 
Bodies near the poles are heavier than those nearer the 
Earth's centre, where the whole force of the Earth's 
attraction is accumulated. They are also heavier, because 
their centrifugal force is less on account of their diurnal 
motions being slower. For both these reasons, bodies 
carried from the poles towards the Equator, gradually 
lose part of their weight. 

Experiments prove that a pendulum which vibrates 
seconds near the poles, vibrates slower near the Equator, 
which shows that it is lighter, or less attracted there. 
To make it oscillate in the same time, it is found neces- 
sary to diminish its length. By comparing the different 
lengths of pendulums vibrating seconds at the equator. 



SEC. III.] ON GRAVITY. 45 

and at London ; it is found that a pendulum must be 
2 5 542 lines* shorter at the Equator than at the poles. 

INTERROGATIONS FOR SECTION THIRD. 

What is Gravity 7 

Do falling bodies strike the surface of the Earth at 
right angles } 

Do falling bodies near the Earth, always direct their 
course to its centre 7 

Where is the centre of gravity situated 7 

When bodies are projected in a right line, what brings 
them to the Earth 7 

If there were no attractive power at the centre of the 
Earth, what would be the consequence were a body so 
projected, and not meeting any resistance from the air 7 

We find that the Moon moves round the Earth in an 
orbit nearly circular. Why is it so 7 

Where is that attractive power situated 7 

Have the other planets attractive powers also 7 

How is it known 7 

Where is the centre of attraction of the Solar System 
placed 7 

How is it known 7 

Do the planets attract the Sun as well as the Sun the 
planets 7 

Has every particle of matter an attractive power 7 

In what proportion does Gravity increase 7 

How far is the moon deflected by Gravity from a tan- 
gent in one minute of time 7 

How far does a falling body descend in one second ? 

In what proportion are the squares of the times of the 
periodical revolutions of all the planets 7 

WTiat will be the form of all planets which revolve on 
their own axis 7 

Why will they be of that form 7 

How is it ascertained to a certainty, that our Earth is 
of that form? 

* A line is l-12th part of an inch. 



46 ON GRAVITY. [SEC. III. 

Why are bodies near the poles heavier than those at 
the Equator ? 

Why is a pendulum vibrating seconds, shorter at the 
Equator than at the poles 7 

What is the length of a pendulum vibrating seconds 
at the Equator 7 

Axis. 39,2 inches 



SEC. IV.] PHENOMENA OF THE HEAVENS, &C. 47 



SECTION FOURTH. 



PHENOMENA OF THE HEAVENS, AS SEEN FROM DIFFER- 
ENT PARTS OF THE EARTH. 

The magnitude of the Earth is only a point when 
compared to the Heavens, and therefore every inhabitant 
upon it, let him be in any place on its surface, sees half 
of the Heavens. The inhabitant on the North Pole of 
the Earth, constantly sees the Northern Hemisphere, 
and having the North Pole of the Heavens directly over 
his head, his horizon coincides with the celestial Equa- 
tor. Therefore all the Stars in the Northern Hemi- 
sphere, between the Equator and the North Pole, appear 
to turn round parallel to the horizon. The Equatorial 
Stars keep in the horizon, and all those in the Southern 
Hemisphere are invisible. The like phenomena are 
seen by an observer at the South Pole. — Hence, under 
either pole, only one half of the Heavens is seen ; for 
those parts which are once visible never set, and those 
which are once invisible never rise. — But the ecliptic or 
orbit, which the Sun appears to describe once a year by 
the annual motion of the Earth, has the half constantly 
above the horizon of the north pole ; and the other half 
always below it. Therefore whilst the Sun describes 
the northern half of the ecliptic, he neither sets to the 
north pole, nor rises to the south ; and whilst he describes 
the southern half, he neither sets to the south pole, nor 
rises to the north. — [See plate 6, figure 2d.] The same 
observations are true with respect to the Moon, with this 
difference only, that as the Sun describes the ecliptic 
but once a year, he is, during half that time, visible tc 
each pole in its turn, and as long invisible. 



48 PHENOMENA OF THE HEAVENS, &C. [sEC. If. 

But, as the Moon goes round the ecliptic in 27 days, 8 
hours, she is only visible during 13 days and 16 hours, 
and as long invisible to each pole by turns. 

All the planets likewise rise and set to the polar regions 
because their orbits are cut obliquely in halves by the 
horizon of the poles. When the Sun arrives at the sign 
Aries, which is on the twentieth of March, he is just 
rising to an observer on the north pole, and setting to 
another on the south pole.* [Plate 6th, figure 2d, spring 
and autumn.] From the Equator, he rises higher and 
higher in every apparent diurnal revolution, till he 
comes to the highest point of the ecliptic on the 21st of 
June, and then he is at his greatest altitude, which is 23 
degrees and 28 minutes ; equal to his greatest north de- 
clination, and from thence he seems gradually to descend 
in every apparent circumvolution till he sets at the sign 
Libra, on the 23d September, and then he goes to exhibit 
the same appearances at the south pole for the other 
half of the year. Hence the Sun's apparent motion 
round the Earth is not in parallel circles, but in spirals, 
such as may be represented by a thread wound round a 
globe from one tropic to the other. If the observer be 
any where on the terrestrial equator, he is in the plane 
of the celestial equator or under the equinox, and the 
axis of the earth is coincident with the plane of his 
horizon extended to the north and south poles of the 
Heavens. As the earth performs her diurnal revolution 
on her axis from west to east, the whole heavens seem 
to turn round the contrary way. It is therefore plain 
that the observer at the equator has the celestial poles 
constantly in his horizon, and that his horizon cuts the 
diurnal paths of all the celestial bodies perpendicularly, 
and in halves. Therefore, the Sun, planets and stars 
rise every day, and ascend perpendicularly above the 
horizon for six hours, and passing over the meridian, 
descend in the same manner, for the six hours folio w- 

* It is therefore evident when the Sun is on the Equator, an ob- 
server placed at each pole, sees about one half of the Sun above the 
horizon, and likewise an observer at the Equator discovers both poles 
in the horizon. 



SEC. IV.] PHENOMENA OF THE HEAVENS, &C. 49 

ing, then set in the horizon and continue 12 hours below 
it ; consequently the days and nights are equally long 
throughout the year. Thus we find, that to an observer 
at either of the poles, one half of the sky is always visi- 
ble, and the other half never seen ; but to an observer 
on the equator, the whole sky is seen every 24 hours. 
From the preceding observations, it is evident that as 
the Sun advances from the equator to the tropic of Can- 
cer, the days continually lengthen, and the nights 
shorten in the northern hemisphere, and the contrary in 
the southern ; and when the Sun descends from the 
equator to the tropic of Capricorn, the days continually 
lengthen in the southern hemisphere, and the nights 
shorten ; and the contrary in the northern. 

The earth's orbit being elliptical, and the Sun con- 
stantly keeping in its lower focus* which is one million 
three hundred and seventy-seven thousand miles from 
the middle point of the longer axis, the earth comes 
twice that distance, or 2,754,000 miles nearer the Sun* 
at one time of the year than at another ; for the Sun 
appearing under a larger anglet in our winter than 
summer, proves that the earth is nearer the Sun in win- 
ter than in summer. The Sun is about 7 days longer in 
the northern hemisphere than in the southern in every 
year ; and as the earth approaches nearer to the Sun, its 
motion is accelerated, and therefore goes over an equal 
space in less time, and as the earth recedes from the Sun, 
its motion is retarded in the same ratio that it was ac- 
celerated when in the southern hemisphere, and con- 
sequently requires a longer time to pass over an equal 
space. 

But here a question actually arises ; why have we 
not the hottest weather when the earth is nearest to the 
Sun 7 In answer, it must be observed, that the eccen- 
tricity of the earth's orbit bears no greater proportion to 
the earth's mean distance from the Sun, than seventeen 

* The Sun is nearest to the Earth when he is on the tropic of Ca- 
pricorn ; farthest from it when he is on the tropic of Cancer. 

t The nearer an object is to the eye, the larger it appears, and 
under the greater angle it is seen. 

5 



50 



SEC. IV. 



bears to a thousand, and therefore this small difference 
of distance cannot occasion any great difference of heat 
or cold. 

But the principal cause of this difference is, that in 
winter, the same rays fall so obliquely upon us, that any 
given number of them is spread over a much greater 
portion of the earth's surface which we inhabit, and 
therefore each point must then have less rays than in 
summer. [See plate 5th, fig. 6.] Also there comes a 
greater degree of cold in the long winter nights, than 
there than can return of heat in so short days, and on 
both these accounts, the cold must increase. But in the 
summer season, the Sun's rays fall more perpendicularly 
upon us, and therefore come with greater force, and in 
greater numbers, on the same place, and by their long 
continuance, a much greater degree of heat is imparted 
by day than can fly off by night. It is for this reason 
that we have a greater degree of heat in the month of 
September, than in the month of March ; the Sun being 
on the equator in both these months, and consequently 
equally distant from the earth. Those parts which are 
once heated, retain the heat for some time, which, with 
the additional heat daily imparted, makes it continue to 
increase, though the Sun declines towards the south, 
and this is the reason why we have greater heat in July 
than in June. Also, we know that the weather is gene- 
rally warmer at 2 o'clock in the afternoon, when the 
Sun has gone towards the west, than at noon, when he 
is on the meridian. Likewise those places which are 
well cooled, require time to be heated again, for the 
Sun's rays do not heat even the surface of any body till 
they have been some time upon it, and therefore January 
is generally colder than December ; although the Sun 
has withdrawn from the winter tropic, and begins to dart 
his beams more perpendicularly upon us. 

It was formerly the opinion of Philosophers, that the 
Sun was an immense mass of flame, and consequently 
the nearer we approached, the greater must be the heat. 
It is stated that the heat of the planet Mercury is seven 



SEC. IV.] PHENOMENA OF THE HEAVENS, &C. 51 

times as great as ours, judging from his nearness to the 
Sun, and likewise that the cold at the planets Jupiter, 
Saturn and Herschel, must be extreme, because they are 
placed at so great a distance from that luminary. 

It is well known, that near the equator, the tops of the 
highest mountains are covered with perpetual snow, and 
that in a less distance than three miles above the sur- 
face of the earth, we come to the region of perpetual con- 
gelation, where neither ice nor snow would ever melt, 
although nearer the Sun than in the plane below. 
Therefore, the distance from the Sun is not the real 
cause of heat or cold. Dark spots have been seen upon 
the Sun's disk, from which it is generally concluded that 
the body of the Sun is dark and opaque, surrounded by a 
luminous atmosphere, which darts its rays with im- 
mense velocity, and by some chymieal operation, per- 
formed in their passage through the atmosphere with 
which this earth is encircled, conveys to us the sensa 
tion of heat. The solar observations of Dr. Wilson, 
first suggested the opinion, that the Sun was an opaque 
and solid body, surrounded with a luminous atmosphere, 
and the telescopes of Dr. Herschel have tended still far- 
ther to establish this opinion. The latter of these astro- 
nomers, therefore imagined, that the functions of the 
Sun, as the source of light, might be performed by the 
agency of the external atmosphere, while the solar nu- 
cleus was reserved, and fitted for the reception of inha- 
bitants. That the Sun may at the same time be the 
source of light and heat, and yet capable of supporting 
animal life is one of those conclusions, which we are 
fond of admitting without hesitation, and to cherish 
with peculiar complacency. The mind is filled with 
admiration of the wisdom of that Benign Benefactor, and 
swells with the most sublime emotions, when it con- 
ceives that apparently the most inaccessible regions of 
creation are peopled with animated beings, and, that 
while the Sun is the fountain of the most destructive of 
all the elements, it is at the same time the abode of life 
and plenty. When the invention of the telescope ena- 



52 PHENOMENA OF THE HEAVENS, &C. [SEC. IV. 

bled Astronomers to detect the striking resemblances be- 
tween the different planets of the system, it was natural 
to conclude, that, as they were composed of similar ma- 
terials, as they revolved around the same centre, and 
were enlightened by similar Moons, they were all in- 
tended by their wise Creator to be the region in which 
he chose to dispense the blessings of existence and in- 
telligence to various orders of animated beings. The 
human mind cheerfully embraced this sublime view of 
creation, and guided by the principle, that nothing was 
made in vain ; man extended his views to the remote 
corners of space, and perceived in every star that spar- 
kles in the sky, the centre of a magnificent system of 
bodies, teeming with life and happiness, and displaying 
fresh instances of the power and beneficence of that Being 
who rolled such stupendous orbs from his creating hand. 
Having thus traversed the illimitable regions of space, 
and considering every world which rolls in the immense 
void as the scene on which the Almighty has exhibited 
his perfections, the mind, unable to command a wider 
range, rests in satisfaction on the faithful analogies which 
it has pursued. 

INTERROGATIONS FOR SECTION FOURTH. 

How much of the Heavens does a spectator see placed 
at the north pole of the earth ? 

What part if placed at the south pole ? 

What part if seen at the equator ? 

When placed at the north pole, how far south in the 
Heavens would his vision extend ? 

If at the south pole, how far north would his vision 
extend ? 

Does any part of the ecliptic, or orbit, which the Sun 
seems to describe once a year, appear above the horizon 
of the north pole? 

What signs of the ecliptic appear above the horizon of 
the north pole ? 

Ans. — Aries, Taurus, Gemini, Cancer, Leo and Virgo ; 
these are called northern signs. 

What signs appear above the horizon of the south pole? 



SEC. IV.] PHENOMENA OF THE HEAVENS, &C. 53 

Arts. — Libra, Scorpio, Sagitarius, Capricornus, Aqua- 
rius, and Pisces. These are called southern signs. . 

When the Sun is in any of the northern signs, does 
he ever set at the north pole, or rise at the south? 

When in the southern signs, does he ever set to the 
south pole, or rise to the north ? 

Is it the same with the Moon ? 

What is the length of the longest days at either of the 
poles ? 

What the length of the longest nights ? 

In what month of the year is the Sun on the tropic of 
Cancer, or highest point of his orbit ? 

In what months on the tropic of Capricorn ? 

In what months on the equator ? 

When the Sun is in the equator, can he be seen at both 
poles ? 

Is the Sun's apparent motion round the earth in cir- 
cles ? 

Are the whole Heavens visible every 24 hours at the 
equator ? 

What is the form of the earth's orbit % 

How many miles is the earth nearer the Sun at one 
time of the year than at another ? 

In what month in the year is it the nearest ? 

In what month farthest oif ? 

How many days is the Sun longer in the northern 
hemisphere, than in the southern in every year ? 

Why is it longer north of the equator than south ? 

Why have we not the warmest weather when nearest 
the Sun? 

What was the opinion of former Philosophers con- 
cerning the Sun ? 

What the opinion of Dr. Herschel ? 

How far above the surface of the earth at the equator, 
is the region of perpetual congelation ? 

What is the prevailing opinion concerning the rays 
of the Sun producing heat ? 

Is the body of the Sun supposed to be inhabited ? 

With what is its dark opaque body surrounded ? 
5* 



54 OF THE MOTIONS OF PLANETS. [SEC. 



SECTION FIFTH. 



PHYSICAL CAUSES OF THE MOTIONS OF THE PLANETS- 

From the uniform projectile motion of bodies in 
straight lines, and the universal power of attraction 
which draws them off from these lines, the curvilinear 
motions of all the planets arise. If a body be projected 
in a right line in open space, and meeting with no re- 
sistance, it would continue for ever to move with the 
same velocity, and in the same direction. But when this 
projectile force is acted upon by any attracting body, with 
a power duly adjusted, and perpendicular to its motion, 
it will then be drawn from a straight line, and forced to 
revolve round the centre of attraction in a circular form. 
But when the projectile force first given exceeds the at 
tracting force, the centrifugal force, or tendency to fly off 
in a tangent, is arrested by the attractive power, and 
therefore its velocity becomes continually more and 
more impeded, until the attractive power has acquired a 
greater influence, and then its motion becomes gradually 
accelerated with a tendency to approach nearer to its 
point of attraction, and consequently the moving body 
forms an elliptical orbit. Let A, a planet, [plate 6th, fig. 
1st,] be projected along the line ABC, meeting with no 
resistance, it would for ever retain the same velocity and 
the same direction. The force which would carry it 
from A to B in a given time, would in an equal time 
carry it from B to C, and from C to D. But if at B it 
fall into the attraction of S, the sun, which should so 
balance the projectile force as to carry it to E at the same 
time that it would by its former motion have arrived at 
C, the planet would now revolve in a circle, B E F. — 



SEC. V.] OF THE MOTIONS OF PLANETS. 55 

But should the attraction of S be more powerful in pro- 
portion to the projectile force, it might bring the planet 
to G, instead of E. or, being stronger, might carry nearer 
the line B S in any given proportion. If carried to G, 
it would revolve in the ellipse B G H. Before it arrives 
at G, and for some distance after, the lines of motion 
caused by the projectile and centripetal forces, form an 
acute angle. The two powers then augment the motion 
caused by each other, and attraction increasing as the 
squares of the distances decrease, the motion of the pla- 
net would be accelerated all the way from B to H. At H 
it would be nearer the centre of attraction than at B by 
twice the eccentricity of its orbit, and being much more 
powerfully attracted, would be drawn to S, were not the 
projectile force also increased. This would now be so 
augmented, that it would carry the planet from H to I 
in the same time that attraction would bring it to H. It 
would then be found at L, and proceed to B, completing 
the revolution. In passing from F to B, the planet would 
be as much retarded in its motion by gravity, as accel- 
erated in its motion from B to F. 

It is readily perceived, that, in two points in the orbit, 
the centripetal and centrifugal forces are equal, and that 
when its motion is both accelerated and retarded by the 
attractive power, it must of necessity pass over equal 
areas in the same space of time. 

As the planets approach nearer the sun, and recede 
farther from him in every revolution, there may be some 
difficulty in conceiving the reason, why the power of 
gravity when it once obtains the victory over the projec- 
tile force, does not bring the planets continually nearer 
the sun in every revolution, till they fall upon and unite 
with him. Or why the projectile force when it once 
gets the better of gravity \ does not continue to carry the 
planets farther from the sun, till it removes them quite 
out of the sphere of his attraction, and go on in straight 
lines for ever afterwards, or when the centripetal and 
centrifugal forces are equal, why it does not commence 
moving off in the form of a perfect circle. By consider- 



56 OF THE MOTIONS OP PLANETS. [SEC. V. 

ing the effects of these powers acting on each other 
according to the preceding description, the difficulty will 
be at once removed. 

A double projectile force will always balance a quad- 
ruple power of gravity, and when a planet is put in mo- 
tion by projectile force, whether the velocity with which 
it moves be rapid or slow, it is continually resisted by the 
attraction of the sun, and consequently moves slower. 
until the power of attraction has gained the ascendency ; 
its motion then becomes accelerated by the centripetal 
power acting on the planet, until its velocity becomes 
equal to the projectile force with which it was first put 
in motion. Therefore it must continue to revolve in an 
elliptical orbit as before stated. And when these two 
forces are equal on the body in motion, they never act 
at right angles, but in such acute angles, that the planet 
is moving with such velocity, that the ascendency is 
instantly obtained. In order to make the projectile force 
balance the gravitating power so exactly, as that the body 
may move in a circle, the projectile velocity of the body 
must be such as it would have acquired by gravity alone, 
in falling through half the radius of the circle. 

By the above mentioned law, bodies will move in ail 
kinds of elliptical orbits whether long or short ; and the 
spaces in which they move in the longer ellipses have 
so much the less projectile force impressed upon them in 
the higher parts of their orbits, and their velocities in 
coming down towards the sun, are so prodigiously in- 
creased by his attraction, that their centrifugal forces in 
the lower parts of their orbits are so great as to overcome 
the sun's attraction there, and cause them to ascend 
again towards the higher parts of their orbits ; during 
which, the sun's attraction acting so contrary to the 
motions of those bodies, causes them to move slower and 
slower until the projectile forces are diminished, almost 
to nothing, and then they are again brought back by the 
sun's attraction as before. 

If the projectile forces of all the planets (and likewise 
those comets, whose mean distances from the sun have 



SEC. V.] OF THE MOTIONS OF PLANETS. 57 

been ascertained,) were destroyed at their mean dis- 
tances from the sun, their gravities would bring them 
down, so that Mercury would fall to the sun in 15 days 
and 13 hours ; Yenus in 39 days and 17 hours ; the 
Earth or Moon in 64 days and 10 hours ; Mars in 121 
days ; Jupiter in 290 days ; Saturn in 767 days ; and 
Herschel in 5406 days ; the nearest comet within the 
orbits of the planets in 13,000days ; the middlemost in 23,- 
000 days, and the outermost in 66,000 days. The Moon 
would fall to the earth in 4 days and 20 hours, Jupiter's 
first moon would fall to him in 7 hours ; his second in 
15 hours ; his third in 30 hours ; and his fourth in 71 
hours. — Saturn's first moon would fall to him in 8 hours ; 
his second in 12 hours ; his 3d in 19 hours ; his 4th in 
68 hours ; his 5th in 336 hours. A stone would fall to 
the earth's centre, if there were a hollow passage, in 21 
minutes and 9 seconds.* 

The rapid motions of the moons of Jupiter and Saturn 
round their primaries, demonstrate that these two pla- 
nets have stronger attractive power than the earth ; for 
the stronger that one body attracts another, the greater 
must be the projectile force, and consequently the force 
pf the other body must be increased to keep it from fall- 
ing to its central planet. Jupiter's second moon is 124,- 
000 miles farther from Jupiter than our moon is from 
us ; and yet this second Moon goes almost 8 times 
around Jupiter, whilst our Moon goes once round the 
( Earth. What a prodigious attractive power must the 
i Sun then have, to draw all the planets and satellites of 
the system towards him, and what an amazing power 
must it have acquired to put all these planets and moons 
into such rapid motions at first. Amazing indeed to 
us, because impossible to be effected by the united 
strength of all the created beings in an unlimited num- 
ber of worlds, but in no wise hard for the Almighty, 
whose Planetarium takes in the whole Universe. 

* The squares of the times, that any planet would fall to the sun, 
are as the cubes of their distances : or multiply the time of a whole 
revolution by ,0176766, the product will be the time in which the 
planet would fall to the sun. 



58 OF THE MOTIONS OF PLANETS. [SEC. V. 

The Sun and planets mutually attract each other, the 
power by which they are thus attracted, is called Gra- 
vity. But whether this power be mechanical or not, is 
very much disputed. Observation proves that the pla- 
nets disturb each other's motions by it, and that it de- 
creases according to the squares of the distances of the 
Sun and planets, as great light which is supposed to be 
material, likewise does. Hence, Gravity should seem 
to arise from the agency of some subtile matter, pressing 
towards the Sun and planets, and acting like all mecha- 
nical causes, by contact. But when we consider that 
the degree or force of Gravity, is exactly in proportion 
to the quantities of matter in those bodies, without any 
regard to their magnitudes or quantities of surface, act- 
ing as freely on their internal as external parts, it ap- 
pears to surpass the powers of mechanism, and to be 
either the immediate agency of the Deity, or affected by 
a law originally established and impressed on all matter 
by him. That the projectile force was at first given by 
the Deity, is evident ; since matter can never put itself 
in motion, and all bodies may be moved in any direc- 
tion whatever, and yet the planets, both primary and 
secondary, move from west to east, in lines nearly co- 
incident, while the Comets move in all directions, and 
in planes very different from each other ; these motions 
can be owing to no mechanical cause or necessity, but 
to the free will and power of an intelligent Being. 

Whatever Gravity be, it is plain that it acts every 
moment of time ; for if its action should cease, the pro- 
jectile force would instantly carry off the planets in 
straight lines from those parts of their orbits where 
Gravity left them. But the planets being once put in 
motion, there is no occasion for any new projectile force, 
unless they meet with some resistance in their orbits, 
nor for any mending hand, unless they disturb each 
other too much by their mutual attraction. 

It is found that there are disturbances among the 
planets in their motions, arising from their mutual at- 
tractions, when they are in the same quarter of the 



SEC. V.] OF THE MOTIONS OF PLANETS. 59 

Heavens, and the best modern observers find that our 
years are not always precisely of the same length. 

If the planets did not mutually attract each other, the 
areas described by them would be exactly proportional to 
the times of description. But observations prove that 
these areas are not in such exact proportions, and are 
most varied when the greatest number of planets are in 
any particular quarter of the Heavens. When any two 
planets are in conjunction, their mutual attractions 
which tend to bring them nearer to each other, draws 
the inferior one a little nearer to him ; by these means, 
the figure of their orbits is somewhat altered, but this 
alteration is too small to be discovered in several ages. 
By the most simple law, the diminution of Gravity, as 
the square of the distance increases, the planets are not 
only retained in their orbits, when whirling with such 
immense velocity around their central Sun ; but an 
eternal stability is insured to the solar system. The 
small derangements which affect the motions of the 
Heavenly bodies, are only apparent, to the eye of the 
Astronomer, and even these, after reaching a certain 
limit, gradually diminish, till the system, regaining its 
balance, returns to that state of harmony and order 
which has preceded the commencement of these secular 
irregularities. Even amidst the changes and lrregulari- 
ties of the system, the general harmony is always ap- 
parent ; and those partial and temporary derangements, 
which to vulgar minds may seem to indicate a progressive 
decay, serve only to evince the stability and permanency 
of the whole. 

In contemplation of such a scene, every unperverted 
mind must be struck with that astonishing wisdom 
which framed the various parts of the Universe, and 
bound them together by one simple, yet infallible law. 
In no part of creation, from the smallest insect to the 
highest seraph, has the Supreme Architect of the Uni- 
verse left himself without a witness ; but it is surely in 
the Heavens above, that the Divine attributes are most 
gloriously displayed. 



60 OP THE MOTIONS OF PLANETS. [sEC. V. 

INTERROGATIONS FOR SECTION FIFTH. 

From what source do the circular motions of the pla- 
nets arise ? 

With what velocity would projected bodies continue 
to move if they met with no resistance ? 

What is meant by the centrifugal force ? 

What is meant by centripetal force ? 

Are the centripetal and centrifugal forces ever equal 
while the planet performs its revolution round the Sun ? 

When the power of Gravity exceeds the projectile 
force, why does it not draw the planets to the Sun ? 

When the projectile, or centrifugal force exceeds the 
attraction, why does it not fly off, and never return ? 

When these forces are equal, why do they not move 
in perfect circles ? 

What will a double projectile force balance ? 

In what case could the projectile be made to balance 
the gravitating power in such manner that the planets 
should move in a perfect circle 7 

If the centrifugal forces should at once be destroyed, 
in what time would each of the planets fall to the Sun ? 

In what time would the Moon fall to the earth ? 

What rule for finding the time in which they would 
fall to the Sun ? 

What do the rapid motions of the Moons of Jupiter 
and Saturn demonstrate ? 

Do the Sun and planets continually attract each other? 

Should gravity instantaneously cease, what would be 
the consequence ? 

Are the motions of the planets continually the same ? 

Do they continue to move exactly in the same path 
at every revolution ? 

By what simple law does gravity diminish ? 



SEC. VI.] ON LIGHT AND AIR. 61 



SECTION SIXTH, 



ON LIGHT AND AIR, 

Light consists of exceedingly small particles of 
matter, issuing from a luminous body, as from a lighted 
candle. Such particles of matter constantly flow in 
every direction. By Dr, Neiwentyt's computation, 
148, 660, 000, 000, 000, 000,000,000,000,000,000,000,000,* 
000,000 particles of light in one second of time flows 
from a candle, which number contains at least 6,337,- 
245,000,000 times the number of grains of sand in the 
whole earth ; supposing 100 grains of sand to be equal 
in length to an inch, and consequently every cubic inch 
of the earth to contain one million of such grains. These 
amazingly small particles, by striking upon our eyes, 
excite in our minds the idea of light, and if they were as 
large as the smallest particle of matter discernible by our 
best microscopes, instead of being serviceable to us, they 
would soon deprive us of sight by the force arising from 
their immense velocity, which is computed at nearly two 
hundred thousand miles in one second.* 

When these small particles flowing from a candle, fall 
upon bodies, and are thereby reflected to our eyes, they 
excite in us the idea of that body, by forming its image 
on the retina.t Since bodies are visible on all sides, light 
must be reflected from them in all directions. A ray of 
light is a continued stream of these particles, flowing from 
any visible body in a straight line. That the rays move 
in straight, and not in crooked lines, (unless they be re- 

* Light passes from the Sun to the Earth in 8 minutes and 7 sec- 
onds, which is 195,072 miles in one second of time. 
f A fine network membrane, in the bottom of the eye. 
6 



62 ON LIGHT AND AIR. [SEC. VI. 

fracted,) is evident from bodies not being visible if we 
endeavour to look at them through the bore of a bended 
pipe ; and from their ceasing to be seen by the interpo- 
sition of other bodies, as the fixed stars, by the interpo- 
sition of the Moon and planets, and the Sun wholly, or 
in part, by the interposition of the Moon, Mercury, or 
Venus. 

There is no physical point, (says Melville,) in the vi- 
sible horizon which does not send rays to every other 
point ; no star in the Heavens which does not send light 
to every other star. The whole horizon is filled with 
rays from every point in it ; and the whole visible Uni- 
verse with a sphere of rays from every star. In short, 
for any thing we know, there are rays of light joining 
every two physical points in the Universe, and that in 
contrary directions, except when opaque bodies inter- 
vene. A ray of light coming from any of the fixed stars 
to the human eye, has to pass in every part of the inter- 
mediate space between the point from which it has been 
projected, and our solar system^ through rays of light 
flowing in all directions from every fixed star in the Uni- 
verse ; and in reaching this earth, it has passed across 
the whole ocean of the solar light, and that which is 
emitted from the planets, satellites, and comets. Yet in 
this course, its progress has not been intercepted. 

The densities and quantities of light, received upon 
any given plane, are diminished in the same proportion, 
as the squares of the distances of that planet from the 
luminous body are increased ; and on the contrary, are 
increased in the same proportion as these squares are 
diminished. 

When a telescope magnifies the disk of the Moon and 
planets, they appeaf more dim than to the bare eye ; 
because the telescope cannot increase the quantity of 
light in the same proportion that it can magnify the sur- 
face, and by spreading the same quantity of light over a 
given surface, it appears more dim than when beheld 
with the naked eye. 

When a ray of light passes out of one medium into 



SEC. VI.] ON LIGHT AND AIR. 63 

another, it is refracted, or turned out of its course more 
or less, as it falls more or less obliquely on the refracting 
surface which divides the two mediums. 

This may be proved by several experiments. In a 
basin, place a piece of money, or any metallic substance, 
and then retire from it till the edge of the basin hides the 
money from your view, then keeping your head steady, 
let another pour water gently into the basin, and as 
it fills with the water, more and more of the substance 
of the bottom will come in sight, and when the basin is 
filled, the substance at the bottom will be full in view, 
and appear as if it was lifted up ; for the ray which was 
straight while the basin was empty, is now bent at the 
surface of the water, and turned out of its natural course 
into an angular direction, and the more dense the me- 
dium is, the more light is reflected in passing through 
it. [Plate 6th, fig. 9th.] 

The earth is surrounded by a thin fluid mass of mat- 
ter, called the Air, or Atmosphere, which gravitates 
to the earth, revolves with it in its diurnal motion, and 
goes with it round the Sun every year. This fluid is of 
an elastic and springy nature, and that part next the 
earth being compressed by the weight of all the air above 
it, is pressed close together, and therefore is the most 
dense at the surface of the earth, and gradually rarer the 
higher you ascend. 

It is well known, that the air near the surface of our 
earth possesses a space about nine hundred times greater 
than water of the same weight, and therefore a cylindric 
column of air nine hundred feet high, is of equal weight 
with a cylinder of water of the same diameter one foot 
high. But a cylinder of air reaching to the top of the 
atmosphere, (45 miles,) is of equal weight with a cylinder 
of water about 33 feet high, and therefore, if from the 
whole cylinder of air, the lower part of nine hundred 
feet high, is taken away, the upper part remaining, will 
be of equal weight with a cylinder of water 32 feet high. 
Wherefore, at the height of nine hundred feet, the weight 
of the incumbent air is less, and consequently the rarity 



64 ON LIGHT AND AIR. [SEC. VI. 

of the compressed air is greater than near the earth's sur- 
face in the ratio of 33 to 32. 

The weight of the air on the earth's surface, is found 
by experiments made with the air pump, and also by the 
quantity of mercury that the atmosphere balances in the 
barometer, in which, at a mean state, the mercury stands 
29 and a half inches high. And if the tube were a 
square inch at the base, and of equal size to the top, it 
would, at that height, contain 29 and a half cubic inches 
of mercury, which is just fifteen pounds ; and conse- 
quently, every square inch of the surface of the earth, 
sustains a weight of 15 pounds; every square foot 2,160 
pounds, at this ratio ; and when the mercury is at that 
height in the barometer, every common sized man sus- 
tains a weight of 32,400 pounds, (the area of the surface 
of his body being about 15 square feet) of air all round; 
for fluids press equally up and down, and on all sides. — 
But because this enormous weight is equal on all sides, 
and counterbalanced by the spring of the air diffused 
through all parts of our bodies, it is not in the least de- 
gree felt by us. 

The state of the air is such many times, that we feel 
ourselves languid and dull, which is generally thought to 
be occasioned by the air's being foggy and heavy about 
us. But at such times, the air is too light. The truth 
of this assertion is known by the sinking of the mercury 
in the barometer, and at these times it is generally found 
that the air has not sufficient strength to bear up the va- 
pours which compose the clouds ; for when it is other- 
wise the clouds ascend high, and the air is more elastic 
and weighty about us, and by these means, it balances 
the internal spring of the air within us, braces the nerves 
and blood vessels, and makes us brisk and lively. 

It is entirely owing to the state of the atmosphere, that 
the Heavens appear bright even in the day time. For, 
without an atmosphere, only that part of the Heavens 
would shine in which the Sun was placed, and if we 
could live without air, and should turn our backs towards 
the Sun, the whole Heavens would appear as dark as in 



SEC. VI.] ON LIGHT AND AIR. 65 

the night ; and the stars would be seen as clearly as in 
the nocturnal sky. In this case we should have no twi- 
light, but a sudden transition from the brightest sunshine 
to the darkness of night, immediately after sunset, and 
from the blackest darkness to the brightest sunshine at 
sun-rising. 

But, by means of the atmosphere, we enjoy the Sun's 
light reflected from the aerial particles for some time 
before he rises, and after he sets. When the earth by its 
rotation has withdrawn our sight from the Sun, the 
atmosphere, (being still higher than we,) has the Sun's 
light imparted to it, which gradually decreases until he 
has descended 18 degrees below the horizon, and then all 
that part of the atmosphere which is above us is dark. 
From the length of twilight, Dr. Reill has calculated the 
height of the atmosphere (so far as it is sufficiently dense 
to reflect any light) to be about forty-four miles high. — 
But it seldom is sufficiently dense at two miles' height 
to bear up the clouds. 

The atmosphere refracts the Sun's rays so as to bring 
him in sight every clear day before he rises in the hori- 
zon, and to keep him in view for some minutes after he 
is really set below it. For, at some times of the year, we 
see the Sun ten minutes longer above the horizon than 
he would be, if there were no refractions, and about six 
minutes every day, at a mean rate. 

INTERROGATIONS FOR SECTION SIXTH. 

Of what does Light consist? 

What would be the consequence, if the particles of 
light were sufficiently large to be discovered by our best 
microscopes ? 

What is meant by the retina ? 

Is light reflected in all directions ? 

What is a ray of light ? 

How is it ascertained that rays of light move in direct 
lines? 

Is light reflected from the Moon to our Earth ? 

Is light sent from one star to another ? 
6* 



66 OP LIGHT AND AIR. [SEC VI. 

In what time does light pass from the Sun to the 
Earth? 

How many miles per second is its velocity ? 

Are the rays of light passing from any luminous body, 
interrupted by those from any other ? 

Are the quantities of light received upon any given 
plane, diminished by being removed at a greater dis- 
tance from that plane ? 

In what proportion are they diminished ? 

What appearance have the moons or planets, when 
their disks are magnified by the aid of a telescope ? 

When a ray of light passes out of one medium into 
another, is it turned out of its former course ? 

If it does, how can it be proved 1 

With what is the earth surrounded ? 

What is the nature of this fluid 7 

Is it capable of being compressed ? 

Is it more dense at the surface of the earth, than some 
distance above 1 

How much heavier is water than atmospheric air ? 

What is the mean height of the atmosphere ? 

By what means is the weight of air at the surface of 
the earth found ? 

What is the weight of air on every square inch ? 

How many pounds on a square foot ? 

How many on the surface of the body of a common 
sized man ? 

In what kind of weather is the air lightest ? 

How is it proved ? 

In what kind the heaviest ? 

At what height may the clouds be borne by the density 
of the atmosphere ? 

How is air rarified ? 

By what is the Sun discovered, when he is in reality 
below the horizon ? 



SEC. VI 1.1 ON DISTANCES OF THE PLANETS, &C. 67 



SECTION SEVENTH. 



TO FIND THE DIAMETER OF THE EARTH, AND THE 
DISTANCES OF THE SUN AND MOON, AND THE DIS- 
TANCES OF ALL THE PLANETS FROM THE SUN.* 

It has been observed that a person at Sea, and ex- 
actly under the equator, discovers both the north and 
south polar stars, just rising in the horizon. Therefore 
as you advance towards either pole, the star will appear 
to rise higher in the horizon, and if you advance ten 
degrees from the equator towards the north pole, the 
polar star will there be ten degrees above the horizon, 
and consequently, if the angle of the elevation of that 
star be taken at any place, the number of degrees of its 
elevation, will be equal to the north latitude of the place 
where the elevation was taken. It has been ascertained 
by actual measurement, that one degree of the surface 
of the earth contains 69 and £ miles nearly. Then, as 
one degree is to 69 and f miles, so is 360 degrees to the 
circumference of the earth. The diameter can then be 
found by the following proportion ; as 355 is to the cir- 
cumference, so is 113 to the diameter. 

Let a large graduated instrument having a more able 
index, with sight holes, be prepared, in such manner, 
that its plane surface may be parallel to the 'plane of the 
equator, and its edge in the meridian, so that when the 
Moon is in the equinox, and on the meridian, she may 
be seen through the sight holes when the edge of the 
moveable index cuts the beginning of the divisions on 
the graduated limb, and when she is so seen, let the pre- 

* Note— Persons unacquainted with Trigonometry, may pass 
over this Section, as they will not be capable of forming correct 
ideas of the methods of finding the accurate distances, and therefore 
must take the Philosopher's word. 



68 ON DISTANCES OF THE PLANETS, &C. [SEC. VII. 

cise time be noted. As the Moon revolves about the 
earth from any meridian to the same again in 24 hours 
and 48 minutes, she will go a fourth part round it in a 
fourth part of that time ; namely six hours and 12 
minutes as seen from the earth's centre or pole : But, as 
seen from the observer's place on the earth's surface, the 
Moon will seem to have gone a quarter round the earth, 
when she comes to the sensible horizon ; for the index 
through the sights of which she is then viewed, will be 
90 degrees from where it was when she was first seen. 
Let the exact moment when she is in, or near the sensi- 
ble horizon be carefully noted,* that it may be known in 
what time she has gone through 90 degrees, as seen by 
the observer on the surface ; subtract this time from the 
aforesaid six hours and 12 minutes, and the remainder 
is the time that she is moving in her orbit, from a tan- 
gent, touching the earth's surface, to a parallel line 
drawn from the earth's centre ; which affords an easy 
matter of finding the Moon's horizontal parallax, which 
is equal to an angle made between the last mentioned 
line, and another drawn from the observer to the centre 
of the Moon, as seen from the centre of the earth or 
poles. — Then, as the afore mentioned remainder is to 
90 degrees, so is 6 hours 12 minutes, to the number of 
degrees which measures the arc ; subtract 90 degrees 
from this arc, and the remainder is the angle under 
which the earth's semi-diameter is seen from the Moon. 
Since all the angles of a right-angled triangle are equal 
to two right angles, or 180 degrees, and the sides of a 
plain triangle are always proportionate to the sines of 
their opposite angles ; say, (by Trigonometry,) as the 
sine of the angle at the Moon is to the earth's semi- 
diameter, so is radius, (or sine of 90 degrees,) to its op- 
posite side ; which is the distance from the observer to 
the Moon, or subtract the angle at the Moon from 90 

* Here proper allowances must be made, for the refraction (being 
about 33 minutes of a degree in the horizon,) will cause the Moon's 
centre to appear 33 minutes above the horizon, when her centre is 
really in it. 



SEC. VII.] ON DISTANCES OF THE PLANETS, &C. 69 

degrees, then say, as the angle at the Moon is to the 
earth's semi-diameter, so is this remainder, to the dis- 
tance from the centre of the earth to the Moon ; which 
comes out at a mean rate 240 thousand miles. The 
Sun's distance from the earth might be found in the 
same manner, if his horizontal parallax were not so 
small as to be hardly perceptible, being 8,632 seconds,* 
while the horizontal parallax of the Moon is 57 minutes 
and 18 seconds. Therefore, to find the distance to the 
Sun, say by single proportion, as the Sun's horizontal 
parallax, (8,632 seconds,) is to the distance that the Moon 
is from the earth, (240,000 miles,) so is the Moon's hori- 
zontal parallax, (57 minutes and 18 seconds,) to the dis- 
tance of the Sun from the earth, which gives in round 
numbers 95 millions of miles. 

The Sun and Moon appear nearly of the same size as 
viewed from the earth, and every person who under- 
stands Trigonometry, knows how their true magnitudes 
may be ascertained from the apparent, when their true 
distances are known. 

Spheres are to each other, as the cubes of their di- 
ameters. Whence, if the Sun be 95 millions of miles 
from the earth, to appear of equal size with the Moon, 
whose distance is only 240 thousand ; he must in solid 
bulk, be 62 millions of times larger than the Moon. 

The horizontal parallaxes are best observed at the 
equator ; Because the heat is so nearly equal everyday, 
that the refractions are almost constantly the same, and 
likewise, because the parallactic angle is greater there ; 
the distance from thence to the earth's axis, being greater 
than upon any parallel of latitude. 

The earth's distance from the Sun being determined ; 

the distances of all the other planets are easily found by 

the following analogy; their periodical revolutions around 

him, being obtained by observation. — As the square of the 

earth's period round the Sun, (a sidereal year) is to the 

cube of its distance from that luminary, so is the square of 

* Ascertained from the transits of Venus across the Sun's disk, in 
She year 1761 and 1769. 



70 ON DISTANCES OF THE PLANETS, &C. [SEC. VII. 

the period of any other planet, to the cube of its distance, 
in such parts or measures, as the earth's distance was 
taken. This proportion gives the relative mean dis- 
tances of the planets from the Sun to the greatest degree 
of exactness. 

The earth's axis produced to the stars, and being car- 
ried parallel to itself during the earth's annual revolu- 
tion, describes a circle in the sphere of the fixed stars, 
equal to the earth's orbit. This orbit, though very large, 
would seem to be no larger than a point, if it were 
viewed from the stars, and consequently, the circle de- 
scribed in the sphere of the stars, by the axis of the 
earth, produced, if viewed from the earth, must appear 
as a point ; its diameter appears too little to be measured 
by observation. 

Dr. Bradley has assured us, that if it had amounted 
to a single second, or two at most, he should have per- 
ceived it in the great number of observations he has 
made ; especially upon Draconis, (a star of the second 
magnitude,) and that it seemed to him very probable 
that the annual parallax of this star, is not so great as a 
single second, and consequently that it is more than four 
hundred thousand times farther from us than the Sun. 
If we suppose that the parallax of the nearest fixed star 
is one second, and that the mean distance of the earth 
from the Sun, is 95 millions of miles, we shall have a 
right-angled triangle whose vertical angle is one second, 
and whose base is 95 millions of miles, to find its side, 
or distance of the star ; which would exceed 20 billions 
of miles, a distance through which light, although travel- 
ling at the rate of two hundred thousand miles in a 
second, could not pass in three years. If the brightest 
star in the Heavens is placed at such an immense dis- 
tance from our system, what an immeasurable interval 
must lie between us and those minute stars, whose light 
is scarcely visible by the aid of the most powerful tele- 
scopes. Many of them are, perhaps so remote, that the 
first beam of light which they sent forth at their cre- 
ation, has not yet arrived within the limits of our sys- 



SEC. VII.] ON DISTANCES OF THE PLANETS, &C. 71 

tern. While other stars which have disappeared, or 
have been destroyed for many centuries, will continue 
to shine in the Heavens till the last ray which they 
emitted, has reached the earth which we inhabit. 

The mean distances of the planets from the Sun, and 
their apparent diameters as seen from that luminary, 
being found, the diameters of all the planets can be as- 
certained by Trigonometry ; thus — Subtract the angle 
or apparent diameter of the planet as seen from the sun, 
from 180 degrees, and half the remainder will be the 
angle at the disk of the planet ; then as the sine of the 
angle at the disk is to the distance of the planet from the 
Sun, so is half the angle at the Sun to the semi-diameter 
of the planet. 

The small apparent motion of the stars discovered by 
that great Astronomer, (Dr. Bradley,) he found to be 
owing to the aberration of their light, which can result 
from no known cause, except that of the earth's annual 
motion : as it agrees so exactly therewith, it proves be- 
yond dispute that the earth has such a motion ; for this 
aberration completes all its various phenomena every 
year, and proves that the velocity of star-light is such as 
carries it through a space equal to the Sun's distance 
from us, in eight minutes and seven seconds of time. 
Hence the velocity of light is about 10,313 times as 
great as the earth's velocity in its orbit, and consequently 
nearly two hundred thousand miles in one second of 
time. 

INTERROGATIONS FOR SECTION SEVENTH. 

From what place can the North and South Poles both 
be seen ? 

Where will they appear ? 

What is an angle of elevation ? 

Suppose the north polar star is elevated 43 degrees 
above the horizon, what is your degree of latitude ? 

How many miles constitute a decree on the surface of 
the earth? 

How is that known 7 



72 ON DISTANCES OF THE PLANETS, &C. [SEC. VII. 

How many degrees in a circle 7 

How is the circumference of the earth found ? 

How the diameter 7 

In what time does the Moon revolve about the earth 
from any meridian to the same again 7 

What is meant by the Moon's horizontal parallax 7 

How is the distance of the Moon found 7 

How is the distance to the Sun found 7 

What proportions do solid bodies bear to each other 7 

When the distance from the Earth to the Sun is found, 
how is the distance from the Sun to the other planets 
ascertained 7 

By what method are the periodical revolutions of the 
planets ascertained 7 

How are the periods ascertained of their revolutions 
on their own axis 7 

Why cannot the exact distance to the fixed stars be 
found 7 

How are the diameters of the planets ascertained 7 

How the diameter of the Sun 7 



SEC. VIII.] OF THE EQUATION OF TlME. 73 



SECTION EIGHTH. 



OF THE EQUATION OF TIME, AND PRECESSION OF THE 
EQUINOXES. 

The stars appear to go around the earth in 23 hours 
56 minutes and 4 seconds, and the Sun in 24 hours. So 
that the stars gain 3 minutes and 56 seconds upon the 
Sun every day ; which amounts to one diurnal revolu- 
tion in a year, or 365 days, as measured by the returns 
of the Sun to the meridian ; there are 366 days as mea- 
sured by the stars returning to it. The former are called 
solar days ; the latter sidereal. 

The earth's motion on its axis being perfectly uni- 
form, and equal at all times of the year ; the sidereal 
days are always precisely of an equal length, and so 
would the solar days be if the earth's orbit were a perfect 
circle, and its axis perpendicular to it. But the earth's 
diurnal motion on an inclined axis, and its annual mo- 
tion in an elliptical orbit, cause the Sun's motion in the 
Heavens to be unequal ; for sometimes he revolves from 
the meridian to the meridian again in somewhat less 
than 24 hours ; shown by a well regulated clock, and 
at other times in somewhat more ; so that the time 
shown by a true going clock, and true sun-dial is never 
the same ; except on the 15th day of April, the 16th of 
June, the 31st of August, and the 24th day of December. 
The clock, if it goes equally true during the whole year, 
will be before the Sun from the 24th of December till the 
15th of April ; from that time till the 16th of June, the 
Sun will be faster than the clock. 

Let S be the sun, (plate 5th fig. 8th) E the earth, AM 
P the earth's orbit, A the aphelion, P the perihelion, the 
line MS the mean prooortional between the semi-axis of 

7 



74 PRECESSION OF THE EQUINOXES. [SEC. VIII. 

the orbit, M a point in the equator represented by the 
external circle of the earth E. Let the spaces AS a M 
S n PS p represent equal areas of the orbit. The arches 
of these by the great law of Kepler represent the earth's 
motion in equal times as a solar day. It is evident that 
the point m when the earth is at a, at n, or at p. must 
pass from m to the line ES to complete a solar day. It 
is also evident that it must pass farther when the earth 
is at p, than when it is at a ; the distance at n being a 
mean between the extremes. A day therefore, measured 
by the Sun, will agree with that shown by a good time- 
keeper, when the earth is at M. At A it will be shorter, 
and at P longer than the true day of the clock. 

The point where the Sun is at his greatest distance 
from the earth is called the Sun's apogee. The point 
where he is at his least distance from the earth is called 
his perigee ; and a straight line drawn through the 
earth's centre from one of those points to the other is 
called the line of the apsides. 

The distance that the Sun has gone in any time from 
his apogee, is called his mean anomaly, and is reckoned 
in signs, degrees, minutes and seconds, allowing 30 de- 
grees to a sign, 

OF THE PRECESSION OF THE EQUINOXES. 

It has been observed, that by the earth's motion on its 
axis, there is more matter accumulated around the equa- 
torial parts than any where else on the surface of the 
earth. The Sun and Moon, by attracting this redun- 
dancy of matter, bring the equator sooner under them in 
every return towards it, than if there were no such ac- 
cumulation. Therefore if the Sun sets out as from any 
star, or other fixed point in the Heavens, the moment 
when he is departing from the equinoctial, or from either 
tropic, he will come to the same equinox or tropic again 
20 minutes and 171 seconds of time, or 50 seconds 
of a degree, before he completes his course so as to arrive 
at the same fixed star, or point from whence he set out. 
For the equinoctial points recede 50 seconds of a degree 



SEC. VIII.] PRECESSION OF THE EQ.UINOXES. 75 

westward every year, contrary to the Sun's annual pro- 
gressive motion. 

When the Sun arrives at the same equinoctial,* or sol- 
stitial point, he finishes what is called the tropical year ; 
which, by observation, is found to contain 365 days, 5 
hours, 48 minutes, and 47 seconds, and when he arrives 
at the same fixed star again, as seen from the earth, he 
completes the sidereal year, which contains 365 days, 6 
hours, 9 minutes, 14 and a half seconds. 

The sidereal year is therefore 30 minutes 17 and a 
half seconds longer than the solar, or tropical year, and 
9 minutes, 14 and a half seconds longer than the Julian 
or civil year, which we state at 365 days, 6 hours. 

As the Sun describes the whole ecliptic, or 360 degrees 
in a tropical year, he moves 59 minutes, 8 seconds of a 
degree every day, at a mean rate ; therefore he will arrive 
at the same equinox, or solstice, when he is 50 seconds 
of a degree short of the same star or fixed point in the 
Heavens, from which he set out in the year before. So 
that with respect to the fixed stars, the Sun and equinoc- 
tial points fall back 30 degrees in 2,160 years, which will 
make the stars appear to have gone 30 degrees forward 
with respect to the signs of the ecliptic in that space of 
time ; for the same signs always keep in the same points 
of the ecliptic, without regard to the constellations, 

The Julian year exceeds the solar by 11 minutes and 
3 seconds, which in 1,438 years amount to eleven days, 
and so much our seasons had fallen back with respect to 
the days of the months since the time of the Nicene 
Council, in A. D. 325, and therefore to bring back all the 
feasts and festivals to the days then settled, it was requi- 
site to suppress 11 nominal days.t And that the same 
seasons might be kept to the same time of the year for 
the future, to leave out the bissextile day in February, at 

* The two opposite points in which the ecliptic crosses the equinox 
are called the equinoctial points, and the two points where the eclip- 
tic touches the tropics, (which are likewise opposite, and 90 degrees 
from the tropic,) are called the solstitial points. 

t The difference in the present century between the old and new 
styles, is twelve days, 



76 PRECESSION OF THE EQ.UINOXES. [SEC. VIM. 

the end of every century, not divisible by four, reckoning 
them only common years, as the 17th, 18th, and 19th 
centuries ; namely, the years 1700, 1800, and 1900, (fee. 
because a day intercalated every fourth year was too 
much, and retaining the bissextile at the end of those 
centuries of years which are divisible by four, as the 
years 1600, 2000, 2400, &c. otherwise in length of time, 
the seasons would be quite reversed with regard to the 
months of the year ; though it would have required near 
23,783 years to have brought about such a total change. 
If the earth had exactly made 365i diurnal revolutions 
on its axis, whilst it revolved from any equinoctial or 
solstitial point to the same again, the civil and solar 
years would always have kept pace together, and the 
style would never have needed any alteration. 

INTERROGATIONS FOR SECTION EIGHTH. 

In what time do the stars appear to go round the 
Earth ? 

In what time does the Sun appear to go round the 
Earth ? 

In what time do the stars gain one revolution 3 

How many days in a solar year ? 

How many in a sidereal ? 

Is the motion of the earth on its own axis uniform at 
all times of the year? 

Are the sidereal days always of the same length ? 

Is the Sun's apparent diameter in the Heavens always 
equal? 

On what days of the year are the Sun and clock 
together ? 

Between what periods will the clock be before the 
Sun ? 

Between what periods will the Sun be before the 
clock? 

What is called the Sun's apogee ? 

What his perigee ? 

What the line of the Apsides ? 

What is meant by the Sun's mean anomaly ? 



SEC. VXII.] PRECESSION OF THE EQUINOXES. 77 

How is his mean anomaly reckoned ? 

What is meant by the precession of the equinoxes ? 

Is there more matter accumulated at the equator than 
at any other part of the earth ? 

What is the cause of such accumulation ? 

What is the Equator ? 

What effect is produced by this accumulation of 
matter ? 

How many seconds of a degree do the equinoctial 
points recede westward every year ? 

What are meant by the equinoctial points ? 

What the solstitial? 

Which is the longest, the sidereal or solar year, and 
how much % 

How many degrees will the equinoctial points fall back 
in 2,160 years? 

Do the same signs always keep in the same point of 
the ecliptie ? 

Which is the longest, the Julian or the solar year, and 
how much ? 

How many days' difference will this make in 1,433 
years? 

In what year of the Christian era was the Council of 
Nice held? 

What centuries were to be leap years ? 

What is the difference between the old and new styles 
in the present century? 



78 OF THE MOON'S PHASES. [SEC. IX. 



SECTION NINTH. 



OF THE MOON'S PHASES. 

By looking at the Moon with an ordinary telescope, 
[see plate 4th, fig. 2d,] we perceive that her surface is 
diversified with long tracts of prodigious high moun- 
tains and dark cavities. This ruggedness of the Moon's 
surface, is of great use to us, by reflecting the Sun's light 
to all sides ; for if the Moon were smooth and polished, 
she could never distribute the Sun's light all around. In 
some positions she would show us his image no larger 
than a point, but with such lustre as would be hurtful to 
our eyes. 

' The Moon's surface being so uneven, many have won- 
dered why her edge does not appear jagged, as well as 
the curve, bounding the light and the dark places. But 
if we consider, that what we call the edge of the Moon's 
disk is not a single line, set round with mountains, in 
which case it would appear irregularly indented, but a 
large zone, having many mountains lying behind each 
other from the observer's eye, we shall find that the moun- 
tains in some rows will be opposite to the vales in others, 
and so fill up the inequalities as to make her appear quite 
round. 

The Moon being an opaque spherical body, (for her 
hills take off no more of her roundness than the inequal- 
ities on the surface of an orange take off from its round- 
ness,) we can only see that part of her enlightened half 
which is towards the earth. Therefore, when she is in 
conjunction with the Sun, her dark half is towards the 
earth, and she disappears ; there being no light on that 
part to render it visible. When she comes to her first 
octant, or has gone over one eighth part of her orbit from 



SEC. IX.] OP THE MOON'S PHASES. 79 

her conjunction, a quarter of her enlightened side is seen 
towards the earth, and she appears horned. When she 
has gone a quarter of her orbit from between the earth 
and Sun, she shows us one half of her enlightened side, 
and then she is said to be a quarter old. When she has 
gone another octant, she shows us more of her enlight- 
ened side, and then she appears gibbous ; and when she 
has gone over half her orbit her whole enlightened side 
is towards the earth, and therefore she appears round : we 
then say it is full Moon, or the Moon is in opposition with 
the Sun. In her third octant, part of her dark side being 
towards the earth, she again appears gibbous, and is on 
the decrease. In her third quarter, she appears half de- 
creased. When in her fourth octant, she again appears 
horned. And after having completed her course from 
the Sun to the Sun again, she disappears, and we say it 
is new Moon. [See plate 4th, fig. 1st.] But when she 
is seen from the Sun, she appears always full. Let S be 
the Sun, [plate 5th, fig. 3d,] E the earth, ABCDEFGH 
the Moon's orbit, the small circle at these letters the Moon 
in different parts of a lunation. The varied appearances 
of the Moon at the earth are represented in the external 
circle at abcdefgh. To understand these requires but a 
slight inspection. 

The Moon's absolute motion from her change to her 
first quarter, is so much slower than the earth's, that she 
falls 240 thousand miles (equal to the semi-diameter of 
her orbit) behind the earth at her first quarter, that is, 
she falls back a space equal to her distance from the earth. 
From that time her motion is gradually accelerated to 
her opposition or full, and then she is come up as far as 
the earth, having regained what she lost in her first quar- 
ter, her motion continues accelerated so as to be just as 
far before the earth as she was behind it at her first quar- 
ter. But from her third quarter her motion is so retarded, 
that she loses as much with respect to the earth, as is 
equal to her distance from it, or to the semi-diameter of 
her orbit, and by that means the earth comes up with 
her, and she is again in conjunction with the Sun, as seen 



80 OP the moon's phases. [sec. IX. 

from the earth. Hence we find that the Moon's absolute 
motion is slower than the earth's from her third quarter 
to her first, and swifter than the earth's from her first 
quarter to her third ; her path being less curved than the 
earth's in the former case, and more in the latter. Yet 
it is still bent the same way towards the Sun ; for if we 
imagine the concavity of the earth's orbit to be measured 
by the length of a perpendicular line let down from the 
earth's place upon a straight line at the full of the Moon, 
and connecting the places of the earth at the end of the 
Moon's first and third quarters ; that length will be about 
640,000 miles, and the Moon when new only approach- 
ing nearer to the Sun by 240,000 miles than the earth. — 
The length of the perpendicular line let down from her 
place at that time upon the same straight line, and which 
shows the concavity of that part of her path, will be 
about 400,000 miles. 

The Moon's path being concave to the Sun through- 
out, demonstrates that her gravity towards the Sun at the 
time of her conjunction, exceeds her gravity towards the 
earth. And if we consider that the quantity of matter 
in the Sun is nearly 230 thousand times as great as the 
quantity of matter in the earth, and that the attraction of 
each body diminishes as the squares of their distances from 
each other increase, we shall soon find that the point of 
equal attraction between the earth and the Sun is about 
70 thousand miles nearer the earth, than the Moon is at 
her change. It may then appear surprising that the 
Moon does not abandon the earth when she is between it 
and the Sun, for she is considerably more attracted by the 
Sun, than by the earth at that time. But this difficulty 
vanishes when we discover that a common impulse on 
any system of bodies affects not their relative motions ; 
but that they will continue to attract, impel, or circulate 
round one another in the same manner as if there were 
no such impulse. The Moon is so near the earth, and 
both of them so far from the Sun, that the attractive power 
of the Sun may be considered as equal on both. There- 
fore the Moon will continue to circulate round the earth 



SEC. IX.] OF THE MOON'S PHASES. 81 

in the same manner as if the Sun did not attract them at 
all. 

OF THE PHENOMENA OF THE HARVEST MOON^ 

It is generally believed that the Moon rises about 50 
minutes later every day than on the preceding ; but this 
is true only to places on the equator. In places of con- 
siderable latitude there is a remarkable difference, espe- 
cially in the time of the autumnal harvest, with which 
farmers were formerly better acquainted than astrono- 
mers. 

In this instance of the Harvest Moon, as in many 
others discoverable by Astronomy, the wisdom and be- 
neficence of the Deity is conspicuous, who really ordered 
the course of the Moon so as to bestow more or less light 
on all parts of the earth, as their several circumstances 
and seasons render it more or less serviceable. 

About the equator, where there is no variety of seasons, 
and the weather seldom changes, except at stated times, 
moonlight is not necessary for gathering the produce of 
the ground, and there the Moon rises about 50 minutes 
later every day or night than on the former. 

In considerable distances from the equator, where the 
weather and seasons are more uncertain, the autumnal 
full Moons rise very soon after sunset for several even- 
ings together. At the polar circles, where the mild season 
is of very short duration, the autumnal full Moon rises 
at sunset from the first to the third quarter : and at the 
poles, where the Sun is during half the year absent, the 
winter full Moons shine constantly without setting from 
the first to the third quarter. 

It is evident that all these phenomena are owing to the 
different angles made by the horizon, and different parts 
of the Moon's orbit, and that the Moon can be full but 
once or twice in a year in those parts of her orbit which 
rise with the least angles. 

The plane of the Equinox is perpendicular to the 
earth's axis, and therefore as the earth turns round in its 
diurnal revolution, all parts of the Equinox make equal 



82 OF THE MOON'S PHASES. [SEC. IX. 

parts with the horizon, both at rising and setting; so that 
equal portions of it always rise or set in equal times. — 
Consequently if the Moon's motions were equable, and in 
the equinox at the rate of 12 degrees and 11 minutes 
from the Sun every day, as it is in her orbit, she would 
rise and set 50 minutes later every day than on the pre* 
ceding ; for 12 degrees 11 minutes of the equator rise or 
set in 50 minutes of time in all latitudes. 

The different parts of the ecliptic, on account of its 
obliquity to the earth's axis, make very different angles 
with the horizon, as they rise or set. These parts or 
signs, which rise with the smallest angles, set with 
the greatest, and rise vice versa. In equal times, 
whenever this angle is least, a greater portion of the 
ecliptic rises than when the angle is larger, as may be 
seen by elevating the pole of a common globe to any con- 
siderable latitude, and then turning it round on its own 
axis in the horizon. Consequently, when the Moon is in 
those signs which rise or set with the smallest angles, 
she rises or sets with the least difference of time, and with 
the greatest difference in those signs which rise or set 
with the greatest angles. On the parallel of London as 
much of the ecliptic rises about Pisces and Aries in two 
hours as the Moon goes through in six days, and there- 
fore, when the Moon is in these signs, she differs but two 
hours in rising for six days together, that is about 20 
minutes later every day or night than on the preceding, 
at a mean rate. But in 14 days afterwards the Moon 
comes to Virgo and Libra, which are the opposite signs 
to Pisces and Aries, and then she differs almost four times 
as much in rising — namely: one hour and about 15 
minutes later every day or night than the former, whilst 
she is in these signs. The annexed table shows the daily 
mean difference of the Moon's rising and setting on the 
parallel of London for 28 days, in which time the Moon 
finishes her period round the ecliptic, and gets 9 degrees 
into the same sign from the beginning of which she set 
out. It appears by the table, that when the Moon is in 
Virgo and Libra, she rises one hour and a quarter later 



SEC. IX.] OF THE MO0N 5 S PHASES. 



83 



every day than she rose on the former, and differs only 
28, 24, 20, 18, or 17 minutes in setting. But when she 
comes to Pisces and Aries, she is only 20 or 17 minutes 
later in rising. 









g. 


p. 








B 1 


Cu 


o 


W 


S 


Is 




d 


88 




3 73 




3 


{3 


ere 
"t 

n 
n 

73 


(B 73 


S3 
ft 


75 


ere' 

73 


CH5 

73' 


S5- 

O OS 

re 


8ff 

ft ere 

CD 








H. M. 


H-M. 








H.M. 


A.M. 


1 


Cancer, 


13 


1 5 


50 


15 




17 


46 


1 5 


2 




26 


1 10 


43 


16 


Aquarius, 


1 


40 


1 8 


3 


Leo, 


10 


1 14 


37 


17 




14 


35 


1 12 


4 




23 


1 17 


32 


18 




27 


30 


1 15 


5 


Virgo, 


6 


1 16 


28 


19 


Pisces, 


10 


25 


1 16 


6 




19 


1 15 


24 


20 




23 


20 


1 17 


7 


Libra, 


2 


1 15 


20 


21 


Aries, 


7 


17 


1 16 


8 




15 


1 15 


18 


22 




20 


17 


1 15 


9 




28 


1 15 


17 


23 


Taurus, 


3 


20 


1 15 


10 


Scorpio, 


12 


1 15 


22 


24 




16 


24 


1 15 


11 




25 


1 14 


30 


25 




29 


30 


1 14 


12 


Sagitarius, 


8 


I 13 


39 


26 


Gemini, 


13 


40 


1 13 


13 




21 


1 10 


47 


27 




26 


56 


1 7 


34 


Capricorn, 


4 


1 4 


56 


28 


Cancer, 


9 


1 00 


1 58 



In the time that the Moon goes round the ecliptic 
from any conjunction or opposition, the earth goes almost 
a sign forward, and therefore the Sun will seem to go as 
far forward in that time ; (namely 27i degrees,) so 
that the Moon must go 27£ degrees more than round, 
and as much farther as the Sun advances in that inter- 
val ; which is 2-rV degrees before she can be in con- 
junction with, or opposite to, the Sun again. Hence, it 
is evident, that there can be but one conjunction, or 
opposition to the Sun and Moon, in any particular part 
of the ecliptic in the course of a year. 

As the Moon can never be full but when she is oppo- 
site to the Sun, and the Sun is never in Virgo and Libra 
only in our autumnal months, it is plain that the Moon 
is never full in the opposite signs, Pisces and Aries, but 
in those two months. Therefore we can have only two 
full Moons in the year, which rise so near the time of 
setting for a week together, as above mentioned. The 



84 OP THE MOON ? S PHASES. [SEC. IX, 

former of these is called the Harvest Moon, the latter the 
Hunter's Moon. 

In northern latitudes, the autumnal full Moons are in 
Pisces and Aries, and the vernal full Moons in Virgo 
and Libra ; in southern latitudes just the reverse, because 
the seasons are contrary. But Virgo and Libra rise at 
as small angles with the horizon in southern latitudes, 
as Pisces and Aries do in the northern ; and therefore 
the Harvest Moons are just as regular on one side of the 
equator, as on the other. 

As these signs which rise with the least angles, set 
with the greatest ; so the several full Moons differ as much 
in their times of rising every night, as the autumnal 
full Moons differ in their times of setting ; and set with 
as little difference as the autumnal full Moons rise ; the 
one being, in all cases, the reverse of the other. 

For the sake of plainness, the Moon has been supposed 
to move in the ecliptic from which the Sun never devi- 
ates. But the orbit in which the Moon really moves is 
different from the ecliptic, one half being elevated b\ 
degrees above it, and the other half, as much depressed 
below. The Moon's orbit therefore, intersects the eclip- 
tic in two points diametrically opposite to each other, 
and these intersections are called the Moon's Nodes. The 
Moon can therefore never be in the ecliplic but when she 
is in either of her nodes, which is at least twice in every 
course, and sometimes thrice. For as the Moon goes 
almost a whole sign more than round her orbit, from 
change to change, if she passes by either node about the 
time of her conjunction, she will pass by the other in 
about fourteen days after, and come round to the former 
node two days again before the next change. That 
node from which the Moon begins to ascend northwardly, 
or above the ecliptic in northern latitudes, is called the 
ascending node, and the other the descending node ; 
because the Moon when she passes by it, descends below 
the ecliptic southward. 

The Moon's oblique motion with regard to the ecliptic, 



SEC. IX.] OF THE~MOON 5 S PHASES. 85 

causes some difference in the times of her rising and 
setting from that which has been already mentioned. 

When she is northward of the ecliptic, she rises 
sooner, and sets later, than if she moved in the ecliptic ; 
and when she is southward of the ecliptic, she rises later 
and sets sooner. This difference is variable even in the 
same signs, because the nodes shift backward about 
19f degrees in the ecliptic every year; and so go 
round it contrary to the order of signs in 18 years and 
225 days. 

When the ascending node is in Aries, the southern 
half of the Moon's orbit makes an angle of 5f degrees 
less with the horizon, than the ecliptic does, when 
Aries rises in northern latitudes; for this reason, the 
Moon rises with less difference of time while she is 
in Pisces and Aries than she would do if she kept in the 
ecliptic. But in 9 years and 112 days afterwards, the 
descending node comes to Aries, and then the Moon's 
orbit makes an angle 5i degrees greater with the 
horizon when Aries rises, than the ecliptic does at 
that time ; which causes the Moon to rise with greater 
difference of times in Pisces and Aries, than if she moved 
in the ecliptic. 

When the ascending node is in Aries, the angle is 
only 9f degrees on the parallel of London when Aries 
rises. But when the descending node comes to Aries, 
the angle is 20 s degrees ; this occasions as great 
difference of the Moon's rising in the same signs every 
9 years, as these would be on two parallels lOf 
degrees from each other, if the Moon's course were in 
the ecliptic. The following table shows how much the 
obliquity of the Moon's orbit affects her rising and set- 
ting on the parallel of London, from the 12th to the 18th 
day of her age, supposing her to be full at the autumnal 
equinoxes, and then either in the ascending node, or 
(highest part of her orbit,) and in the descending node, 
(or lowest part of her orbit.) M signifies morning ; A 
afternoon, and the line at the foot of the table shows a 
week's difference in rising and setting. 



86 



OF THE MOON'S PHASES. 



[sec. IX. 





Full in her 


Full in the highest 


Full in her 


Full in the lowest 




ascending node. 


part of her orbit. 


descending node. 


part of her orbit 


^HrSmb at! Sets at 
r ' |h m|h m 


Rises at Sets at Rises at Sets at 


Rises at Sets at 


H M|H M|H M|H M 


H M 1 H M 


12 ig A 15 


3 M 20 4 A 30 


3 M 15 


4 A 32 


3 M 40 


5 A 16 


3 M 


13 5 32 


4 25 


4 50 


4 45 


5 15 


4 20 


6 


4 15 


14 5 48 


5 30 


5 15 


6 


5 45 


5 40 


6 20 


5 28 


15 6 5 


7 


5 42 


7 20 


6 15 


6 56 


6 45 


6 32 


16 6 20 


8 15 


6 2 


8 35j6 46 


8 


7 8 


7 45 


17 6 36 


9 12 


6 26 


9 45 7 18 


9 15 


7 30 


9 15 


18 |6 54 


10 30 


7 


10 4018 


10 20 


7 52 


10 


differ- 


1 3917 10 


2 30 


7 25 i 3 28 


6 40 2 36 7 


ence. 






1 









This table was not computed, but only estimated, as 
near as could be done from a common globe, on which 
the Moon's orbit was delineated with a black lead pencil. 

As there is a complete revolution of the nodes in 18f 
years, there must be a regular period of all the varieties 
which can happen in the rising and setting of the Moon 
during that time. 

At the polar circles, when the Sun touches the sum- 
mer tropic, he continues twenty -four hours above the 
horizon ; and the like number below it, when he touches 
the winter tropic. For the same reason, the full Moon 
being as high in the ecliptic as the summer's Sun, must 
therefore continue as long above the horizon ; and the 
summer full Moon being as low in the ecliptic as the 
winter Sun, can no more rise than he does. But these 
are only the two full Moons which happen about the 
tropics, for all the others rise and set. In summer, the 
full Moons are low, and their stay is short above the 
horizon ; then the nights are short, and we have the least 
occasion for moonlight. In winter, the full Moons run 
high, and they stay long above the horizon when the 
nights are long, and we need the greatest quantity of her 
reflected light. 

At the poles, one half of the ecliptic never sets, and 
the other half never rises ; and therefore as the Sun is 
always half a year in describing one half of the ecliptic, 
and as long in going through the other half, it is natural 



SEC. IX.] OF THE MOON'S PHASES. 87 

to imagine that the Sun continues half a year together 
above the horizon of each pole in its turn, and as long 
below it ; rising to one pole when he sets to the other. 
This would be exactly the case, if there were no refrac- 
tion. But by the refraction of the Sun's rays, occasioned 
by the atmosphere, he becomes visible some days sooner, 
and continues some days longer in sight, than he would 
otherwise do : so, that he appears above the horizon of 
either pole, before he has got below the horizon of the other. 
And as he never goes more than 23 degrees and 28 
minutes below the horizon of the poles, they have very 
little dark night ; twilight being there, as well as at all 
other places, till the Sun be 18 degrees below the horizon. 
The full Moon being always opposite to the Sun, can 
never be seen while the Sun is above the horizon, except 
when the Moon falls in the northern half of her orbit ; 
for when any point of the ecliptic rises, the opposite 
point sets. Therefore, as the Sun is above the horizon 
of the north pore from the 20th of March, till the 23d 
September ; it is plain that the Moon when full, being 
opposite to the Sun, must be below the horizon during 
that half of the year. But when the Sun is in the 
southern half of the ecliptic, he never rises to the north 
pole ; during this half of the year every full Moon hap- 
pens in some part of the northern half of the ecliptic 
which never sets. Consequently, as the polar inhabit- 
ants never see the full Moon in summer, they have her 
always in the winter ; before, at, and after the full, shin- 
ing during 14 of our days and nights. And when the 
Sun is at his greatest distance below the horizon, being 
then in Capricorn, the Moon is at her first quarter in 
Aries, full in Cancer, and at her third quarter in Libra. 
And as the beginning of Aries is the rising point of the 
ecliptic, Cancer the highest, and Libra the setting point, 
the Moon rises at her first quarter in Aries, is most ele- 
vated above the horizon, and full in Cancer, and sets at 
the beginning of Libra in her third quarter, having con- 
tinued visible for 14 diurnal rotations of the earth. Thus 
the poles are supplied one half of the winter time with 



88 OF THE MOON'S PHASES. [SEC. IX. 

constant moonlight, in the absence of the Sun, and only 
lose sight of the Moon from her third to her first quarter, 
while she gives but very little light, and could be but of 
little, and sometimes of no service to them. 

INTERROGATIONS FOR SECTION NINTH. 

What is understood by the Moon's Phases ? 

What is discovered by observing the Moon with a 
telescope 7 

Of what use is the ruggedness to us 7 

If the surface of the Moon is uneven, why does it not 
so appear when viewed by the eye only 7 

What is the Moon 7 

What part of the Moon do we discover 7 

When is she said to be in conjunction with the Sun 7 

When she is in her first octant, how much of her 
enlightened side is visible 7 

How much of her enlightened side does she show in 
her first quarter 7 

When she is gone half around her orbit, how does she 
appear 7 

How does she appear when viewed from the Sun 7 

Are the Moon's motions faster or slower than the 
earth's from her change to her first quarter 7 

How far does she fall behind the earth 7 

From her first quarter to her full, which moves with 
the greatest rapidity 7 

Which from the full to her third quarter ? 

Which from the third quarter to the change 7 

Is the gravity of the Moon at any time greater towards 
the Sun, than towards the earth, and at what time 7 

How much greater is the quantity of matter in the 
Sun, than in the earth 7 

In what proportion does the attraction of each body 
diminish 7 

How far from the earth is the point of equal attrac- 
tion between the earth and the Sun 7 

Why does not the Moon leave the earth and go to the 
Sun? 



SEC. IX.] OF THE MOON'S PHASES. 89 

What is understood by the Harvest Moon ? 

How many minutes later at the equator does the Moon 
rise every day, than on the preceding ? 

Is there any material difference in high northern or 
southern latitudes ? 

At what time in northern latitudes, does the full Moon 
rise? 

How many days together does the Moon in such cases 
rise at nearly the same time ? 

What is the cause of this small difference ? 

How far does the earth advance in her orbit, while 
the Moon goes round the ecliptic ? 

How many conjunctions and oppositions of the Sun 
and Moon can take place in any particular part of the 
ecliptic, in the course of a year ? 

How many full Moons in the course of a year, that 
rise with so little difference near the time of Sun-setting ? 

Does this singularity appear in southern latitudes, as 
well as in northern ? 

Does the Moon's orbit lie exactly in the ecliptic ? 

Does the Moon's orbit intersect the ecliptic ? 

What is understood by the Moon's Nodes ? 

How many times from change to change, is the Moon 
in her nodes ? 
'. Which is called the ascending node ? 

Which is called the descending node ? 

How many degrees are they asunder ? 

How much does these nodes shift in the course of a year? 

Which way do they shift ? 

In what length of time do they go around the ecliptic ? 

How many degrees can the Sun go below the horizon 
of the poles ? 

How many degrees must the Sun be below the hori- 
zon before the twilight is wholly gone ? 

Is the full Moon in the summer season ever seen at 
the north pole ? 

Is it continually seen in winter, from the first to her 
third quarter ? 

Is it the same at the south pole ? 



90 ON TIDES. [SEC. X. 



SECTION TENTH. 



ON TIDES. 

The cause of the tides was first discovered by Kepler, 
who thus explains it. The orb of the attracting power 
(which is in the Moon) is extended as far as the earth, 
and draws the waters under the torrid zone ; acting upon 
places where it is vertical — insensibly on confined seas 
and bays, but sensibly on the ocean, whose beds are 
larger, and whose waters have the liberty of reciprocation, 
that is of rising and falling. And in the 70th page of 
his lunar Astronomy he says : But the cause of the tides 
of the sea, appears to be the bodies of the Sun and Moon, 
drawing the waters of the sea. 

This hint being given, the immortal Sir Isaac New- 
ton improved it, and wrote so amply on the subject as 
to make the theory of Tides in a manner quite his own, 
by discovering the cause of their rising on the side of 
the earth opposite to the Moon. For Kepler believed 
that the presence of the Moon occasioned an impulse 
which caused another in her absence. 

It has been already mentioned, that the power of 
gravity diminishes as the square of the distance increases, 
and therefore the waters on the side of the earth, next 
the Moon, are more attracted than the central parts of 
the earth by the Moon, and the central parts are more 
attracted by her, than the waters on the opposite side of 
the earth. Therefore the distance between the earth's 
centre, and the water, or its surface, will be increased. 
If the attraction be unequal, then that body which is 
most strongly attracted will move with greater rapidity, 
and this will increase its distance from the other body. 
Consequently, the unequal attraction of one part of the 



SEC. X.] ON TIDES. 91 

terraqueous globe more forcibly than the other, may be 
considered as the true cause of the tides. 

As this explanation of the ebbing and flowing of the 
sea is deduced from the earth's constantly falling toward 
the Moon by the power of gravity, some may find a 
difficulty in conceiving how this is possible when the 
Moon is full, or in opposition to the Sun ; since the 
earth revolves about the Sun, and must continually fall 
towards it ; and therefore cannot fall contrary ways at 
the same time : or if the earth is constantly falling 
towards the Moon, they must come together at last. To 
remove this difficulty, let it be considered that it is not 
the centre of the earth that describes the annual orbit 
round the Sun, but the common centre* of gravity of 
the earth and Moon together ; and that while the earth 
is moving round the Sun. it also describes a circle 
around that centre of gravity, going as many times 
around it in one revolution about the Sun, as there are 
lunations, or courses of the Moon around the earth ; is 
constantly falling towards the Moon from a tangent to 
the circle it describes, around the said centre of gravity. 

The influence of the Sun in raising the tides, is but 
small in comparison of the Moon's : though the earth's 
diameter bears a considerable proportion to its distance 
from the Moon, it is next to nothing when compared to its 
distance from the Sun. Therefore, the difference of the 
Sun's attraction on the sides of the earth, under, and op- 
posite to him, is much less than the difference of the 
Moon's attraction on the sides of the earth under, and op- 
posite to her ; therefore the Moon must raise the tides 
much higher than they can be raised by the Sun. On 
this theory, (so far as it has been explained,) the tides 
ought to be the highest directly under, and opposite to 

* This centre is as much nearer the earth's centre than the 
Moon's, as the earth is heavier, or contains a greater quantity of 
matter than the Moon, which is about 40 times. If both bodies were 
suspended from it, they would hang in equilibrio. Therefore 
divide the Moon's distance from the earth's centre, (240,000 miles,) 
by 40, and the quotient will be the distance from the centre of the 
earth to the centre of gravity, which is 6,000 miles, or 2,000 from 
the earth's surface. 



92 ON TIDES. [SEC. X. 

the Moon ; that is, when the Moon is due north or south. 
But we find in open seas, where the water flows freely, 
the Moon is generally past the north and south meridian 
when it is high water. The reason would be obvious, were 
the Moon's attraction to cease wholly when she was past 
the meridian, yet the motion of ascent communicated to 
the water before that time, would make it continue to rise 
for some time afterward, much more must it continue to 
rise when the attraction is only diminished. A little 
impulse given to a moving ball, will cause it to move 
farther than it otherwise would have done. Or, as ex- 
perience shows that the weather in summer is warmer 
at 2 o'clock in the afternoon than when the Sun is on the 
meridian, because of the increase made to the heat 
already imparted. 

The tides do not always answer to the same distance 
of the Moon from the meridian at the same places, but 
are variously affected by the action of the Sun, which 
brings them on sooner when the Moon is in her first and 
third quarters, and keeps them back later when she is in 
her second and fourth ; because, in the one case, the tide 
raised by the sun alone, would be earlier than the tide 
raised by the Moon, and in the other case later. 

The Moon goes round the earth in an elliptical orbit, 
and therefore in every lunar month she approaches nearer 
to the earth than her mean distance, and recedes farther 
from it. When she is nearest, she attracts strongest, and 
so raises the tides most ; the contrary happens when she 
is farthest, because of her weaker attraction. 

When both luminaries are in the equator, and the 
Moon in her perigee, (or least distance from the earth,) 
she raises the tides highest of all ; especially at her con- 
junction and opposition, both because the equatorial parts 
have the greatest centrifugal force from their describing 
the largest circle, and from the concurring actions of the 
Sun and Moon. [See plate 6th, fig. 4th.] At the change, 
the attractive forces of the Sun and Moon being united, 
they diminish the gravity of the waters under the Moon, 
and their gravity on the opposite side is diminished by 



SEC. X.] ON TIDES. 93 

means of a greater centrifugal force. At the full, while 
the Moon raises the tide under, and opposite to her, the 
Sun acting in the same line, raises the tide under, and 
opposite to him ; whence their conjoint effect is the same 
as at the change, and in both cases occasion what is 
called Spring Tides. [Fig. 7th, plate 6th.] But at the 
quarters, the Sun's action diminishes the action of the 
Moon on the waters, so that they rise a little under, and 
opposite to the Sun, and full as much under, and oppo- 
site to the Moon, making what we call neap tides ; 
because the Sun and Moon then act crosswise to each 
other. [Fig. 7th, plate 6th.] But, strictly speaking, 
these tides happen not till some time after, because in 
this, as in other cases, the actions do not produce the 
greatest effect when they are at the strongest, but some- 
time afterward. The Sun being nearer the earth in 
winter than in summer, is of course nearer to it in Feb- 
ruary and October than in March and September, and 
therefore the greatest tides happen not till some time 
after the autumnal equinox : and return a little before the 
vernal. The Sea being thus put in motion, would con- 
tinue to ebb and flow for several times, though the Sun 
and Moon should be annihilated, or their influence cease. 
When the Moon is in the equator, the tides are equally 
high in both parts of the lunar day, or time of the Moon's 
revolving from the meridian to the meridian again, 
which is 24 hours and 50 minutes. But, as the Moon 
declines from the equator towards either pole, the tides 
are alternately higher and lower at places having north 
or south latitude. One of the highest elevations, (which 
is that under the Moon,) follows her towards the pole to 
which she is nearest, and the other declines towards the 
opposite pole ; each elevation describing parallels as far 
distant from the equator on opposite sides, as the Moon 
declines from it to either side, and consequently, the 
parallels described by these elevations of the water, are 
twice as many degrees from each other as the Moon is 
from the equator, increasing their distance as the Moon 
increases her declination, till it be at the greatest : when 



94 ON TIDES. [SEC. X. 

the said parallels are at a mean state 47 degrees asunder, 
and on that day the tides are most unequal in their 
heights. As the Moon returns towards the equator, the 
parallels described by the opposite elevations approach 
toward each other until the Moon comes to the equator, 
and then they coincide. As the Moon declines toward 
the opposite pole at equal distances, each elevation de- 
scribes the same parallel in the other part of the lunar 
day, which its opposite elevations described before. — 
While the Moon has north declination, the greatest tides 
in the northern hemisphere, are when she is above the 
horizon, [see plate 6th, fig. 5th,] and the reverse when 
her declination is south. [See plate 6th, fig. 4th.] 

Thus it appears, that as the tides are governed by the 
Moon, they must tum*on the axis of the Moon's orbit, 
which is inclined 23 degrees and 28 minutes to the earth's 
axis at a mean state, and therefore the poles of the tides 
must be so many degrees from the poles of the earth, or 
in opposite points of the polar circles, going around them 
in every revolution of the Moon from any meridian to 
the same again. [Plate 6th, fig. 3d.] 

It is not, however, to be doubted, but that the quick 
rotation of the earth on his axis, brings the poles of the 
tides nearer to the poles of the world than they would 
be if the earth were at rest, and the Moon revolved about it 
only once a month, otherwise the tides would be more 
unequal in their heights, and times of their returns, than 
we find they are. But how near the earth's rotation may 
bring the poles of its axis and those of the tides together, 
or how far the preceding tides may affect those that fol- 
low, so as to make them keep up nearly to the same 
heights and times of ebbing and flowing, is a problem 
more fit to be solved by observation than theory. 

In open seas, the tides rise but to very small heights 
in proportion to what they do in broad rivers, whose wa- 
ters empty in the direction of the stream of tide : — For, 
in channels growing narrower gradually, the water is 
accumulated by the opposition of the contracting bank. 
The tides are so retarded in their passage through dif~ 



SEC. X.] ON TIDES. 95 

ferent shoals and channels, and otherwise so variously 
affected by striking against capes and headlands, that to 
different places, they happen at all distances of the Moon 
from the meridian, and consequently at all hours of the 
lunar day. 

Air* being lighter than water, and the surface of the 
atmosphere nearer to the Moon than the surface of the 
sea, it cannot be doubted that the Moon raises much 
higher tides in the air than in the sea. 

INTERROGATIONS FOR SECTION TENTH. 

By whom was the cause of the tides first discovered 7 

How does he explain it 7 

Who improved the idea of Kepler 7 

By what does he consider the waters to be attracted 7 

Why are the waters on the side of the earth next to 
the Moon, more attracted than the central parts 7 

Why are the central parts more attracted, than the 
waters on the opposite side 7 

From what source is this explanation deduced 7 

By what power is the earth constantly falling towards 
the Moon, and the Moon towards the earth 7 

If this be actually the case, why do they not come 
together 7 

Is it the centre of the earth that describes the annual 
orbit round the Sun 7 

Where is the centre of gravity between the earth and 
Moon? 

How much more matter does the earth contain, than 
the Moon 7 

What is the centre of gravity between the two bodies 7 

How is it found 7 

* In a register of the barometer kept for 30 years, the Professor 
Toaldo of Padua, added together all the heights of the mercury, 
when the Moon was in syzy gy, when she was in quadrature, and 
when she was in the apogeal and perigeal points of her orbit. The 
apogeal exceeded the perigeal heights by 14 inches, and the heights 
in syzygy exceeded those in quadrature by 11 inches. The difference 
1 n these heights is sufficiently great to show that the air is accumu- 
lated and compressed by the attraction of the Moon. 



96 ON TIDES. [SEC. X. 

Which has the greatest influence in raising tides, the 
Sun or Moon 2 

Are the tides at the highest when the moon is due 
lorth, or south 1 

What is the reason ? 

Do the tides always answer to the same distance of 
the Moon from the meridian at the same places ? 

Does the Moon approach nearer, and recede farther 
from the earth in each of her revolutions ? 

At what time does she attract the earth most ? 

At what time does she attract it the least ? 

In what positions are the Sun and Moon when the 
highest tides are raised ? 

"What are spring-tides ? 

What are neap4ides 1 

In what manner do the attractions of the Sun and 
Moon act on each other, to produce spring-tides ? 

In what manner to produce neap-tides 1 

Where is the Moon when the tides are equally high in 
both parts of the lunar day ? 

What is understood by the lunar day ? 

What is the length of the lunar day ? 

At what time are the tides most unequal ? 

In which hemisphere are the highest tides, when the 
Moon has north declination ? 

Which when in her south declination ? 

Do the tides rise very high in open seas ? 

Are the tides ever retarded in their passage 1 

What retards them ? 

What are aerial tides ? 

How were they discovered, and by whom ? 



SEC. XI.] ASTRONOMICAL PROBLEMS. 97 



SECTION ELEVENTH. 



ASTRONOMICAL PROBLEMS. 

PROBLEM I. 

To convert time into degrees, minutes, fyc. 
Rule. — As one hour is to 15 degrees, so is the time 
given to the answer. 

EXAMPLES. 

1 . How many degrees are equal to 8 hours, 20 minutes, 
and 30 seconds 7 

2d. The Sun passes the meridian of Detroit 1 hour 
19 minutes after 12 o'clock, noon, at Boston — How far are 
those places asunder 7 

PROBLEM II. 

To convert degrees, minutes, fyc. into time. 
Ride. — As 15 degrees are to an hour, so are the num- 
ber of degrees given to the time. 

EXAMPLES. 

1. The apparent distance of Venus from the Sun, can 
never be above 50 degrees, and when at that distance, 
how long does she rise before the Sun, or set after 
him? 

2. The greatest elongation of Mercury is said to be 28 
degrees, 20 minutes and 19 seconds — How long can he 
set after the Sun, when an evening star 7 

PROBLEM III. 

The diurnal arc of the Sun, or of any planet being 
given, to find the time of the rising or setting of the Sun. 

Rule.— Bring the diurnal arc into time by Problem 
2d. Divide this time by two, and the quotient will be 
the time at which the Sun sets. Take this time from 

9 



98 ASTRONOMICAL PROBLEMS. [sEC. XI. 

12 hours, and 
the Sun rises. 



12 hours, and the remainder will be the time at which 



EXAMPLES. 

1. Suppose the Sun's diurnal are be 174 degrees and 
thirty minutes, at what time does he rise and set? 
Ans. 5 hours 49 minutes, the time of the Sun's set- 
ting, and he rises at 6 hours and 11 minutes. 

2. The diurnal arc of Venus is found to be 96 degrees 
and 44 minutes — At what hours does the Sun rise, and 
when does he set ? 

3. The diurnal arc of Mars, is 198 degrees, 14 minutes 
and 50 seconds. 

The diurnal arc of Jupiter, is 201 degrees, 33 minutes 
and 16 seconds. 

The diurnal arc of Saturn, is 196 degrees, and 14 min- 
utes : and the diurnal arc of Herschel is 213 degrees, 
41 minutes, and 58 seconds ; when, according to the 
above-mentioned numbers, does the Sun rise and set? 

PROBLEM IV. 

The time which the Sun, or any planet remains above 
the horizon being given, to find the length of his diurnal, 
or nocturnal arc. 

Rule. — Divide the given time by two, and the quo- 
tient will be the time of the Sun's setting. Take this 
time from 12 hours, and the remainder will be the time 
of his rising. Multiply the given time by 15 degrees, 
and the product will give the Sun's, or planet's diurnal 
arc ; — this subtracted from 360 degrees, will leave the 
nocturnal arc. 

examples. 

1. On the fourth of July, the Sun rose at 43 minutes 
past 5 o'clock — At what time did he set on that day, 
and what was the length of his diurnal arc ? 

2. September 7th, 1825, the Sun rose at 5 o'clock and 
52 minutes — At what time did he set, and what were the 
dimensions of both arcs? 



SEC. XI.] ASTRONOMICAL PROBLEMS. 99 

PROBLEM V. 

To find the time which elapses between two conjunc- 
tions, or two oppositions, or between one conjunction, 
and one opposition of any two planets. 

Rule. — Find the difference between the given daily 
motions of the two given planets, as given in the follow- 
ing table of the daily motions, then say, as the difference 
of their daily motions, is to one day, so is 360 degrees, to 
the difference in the times of the two conjunctions, or 
oppositions required. But for one conjunction, and one 
opposition, or, for a superior and an inferior conjunction ; 
say as the difference of their daily motions is to one day, 
so is 180 degrees to the time, which elapses between a 
conjunction, and an opposition of the two given planets. 

TABLE. 

D. 

Mercury's daily motion is ... . 4,0928 degrees. 

Venus's do. do. 1,6021 

The Earth's do. do 0,9856 

Mars's do. do. 0,5240 

Jupiter's do. do 0,0831 

Saturn's do. do 0,0335 

HerscheFs do. do 0,0118 

EXAMPLES. 

1. How many days elapse between a conjunction, and 
an opposition of Mercury and Yenus ? 

Thus Mercury's daily motion is 4,0928 degrees, less 
the daily motion of Yenus, 1,6021 — 2,4907 degrees • 
then as 2,4907 d : 1 day : : 180 d : 72,25 days, the time 
required. 

2. How many days is Yenus a morning and an even- 
ing star, alternately to the earth ? 

3. How many days is Jupiter a morning and even- 
ing star, alternately to the earth ? 

4. How many days is Mercury east, and how many 
west of the Sun to us ? 

The heliocentric longitude of any two planets being 



100 ASTRONOMICAL PROBLEMS. [SEC. XL 

PROBLEM VI. 

given, to find when they will be in heliocentric con- 
junction. 

Rule. — Subtract the given longitude of the planet 
nearest the Sun, from that of the planet farthest from 
him, if practicable, but if not, add to the latter 360 de- 
grees, and then subtract, say, as the difference of the 
daily motions of the given planets is to one day, so is the 
difference of their longitudes to the time when the given 
planets will be in conjunction. 

EXAMPLES. 

1. At what time were Mars and Venus in conjunction, 
after the first of January, 1823 ? 

Venus's daily motion is 1,6021 — 0,5240 = 1,0781; 
Mars's longitude for January 1st, 1832, was 311 degrees 
and 41 minutes : less by 285 degrees, 16 minutes ; the 
longitude of Venus at the same time = 26 degrees and 
25 minutes. Then as 1,0781 : 1 day :: 26 d. 25 m. = 
24,5, or January 25th, 1823. 

TABLE. 

The Sun's geocentric longitude for January 

1st, 1823, was, 280° 29' 

The heliocentric longitude of Mercury, was 277 25 

That of Venus, was 285 16 

The Earth, 100 20 

That of Mars, 311 41 

Jupiter, 64 51 

Saturn, 38 56 

Herschel, 277 30 

On what day of the year, 1823, was Venus in con- 
junction with the earth ? 

3. When was Jupiter in conjunction with the earth in 
the year 1824? 

4. When were Venus and Jupiter in conjunction, in 
1825? 

5. On what day in the year 1832, did Jupiter rise at 
the moment the Sun arose ? 



SEC. XI.] ASTRONOMICAL PROBLEMS 101 

PROBLEM VII. 

When the heliocentric longitude of any planet, for any- 
given day is known, to find it for any required day. 

Rule. — Find the number of days between the given, 
and required day : then as one day is to the given 
planet's daily motion, so are the days so found, to the 
distance which the planet has revolved during that time. 
Add this distance to the planet's known longitude, and 
the sum, if less than 360 degrees, will be the longitude 
for the required day, but if more than 360 degrees, then 
subtract 360 degrees from it, and the remainder will be 
the true longitude. 

EXAMPLES. 

1. On the first of January, 1823, the heliocentric lon- 
gitude of Venus was 285 degrees, 16 minutes — What 
was it on the 4th of July, in the same year ? Ans. 220 
degrees and three minutes. 

2. On the first of January, 1823, the earth's longitude 
was 100 degrees and 20 minutes — What was its longi- 
tude on the 4th of July, 1825 ? 

PROBLEM VIII. 

To determine whether Venus or Jupiter will be the 
morning or evening star on any given day. 

Rule. — Find the longitude of Venus and the longi- 
tude of the earth for the given day. If the difference in 
longitude, counting from the earth's place eastward, be 
less than 180 degrees, Venus will be east of the Sun, and 
consequently evening star: but if that difference be 
greater than 180 degrees, she will be west of the Sun, 
and therefore morning star. 

EXAMPLES. 

On the 4th of July, 1823 > was Venus a morning or an 
evening star 7 

The longitude of Venus on the given day, will be 
found by Problem 7th, to be 220 degrees and 3 minutes, 
and the earth's longitude, for the same day by the same 
Problem. 281 degrees and 41 minutes : the difference = 

9* 



102 ASTRONOMICAL PROBLEMS. [SEC XI 

61 degrees and 38 minutes ; this difference being less 
than 180 degrees, shows that Yenus is east of the Sun ; 
and consequently an evening star. 

Did Jupiter rise before, or after the Sun, July 4th, 1832? 

How many days in succession, can Venus be a morn- 
ing or an evening star ? 

How many days in succession, can Jupiter be a morn- 
ing or an evening star ? 

PROBLEM IX. 

To determine the day on which any particular planet 
shall have a given longitude. 

Rule. — Subtract the longitude of the given planet 
found in the preceding table, from the given longitude, 
if practicable ; but if the longitude of the planet found in 
the table, be greater than the given longitude, increase 
the latter by 360 degrees, and then subtract ; divide the 
remainder by the planet's daily motion, as recorded in 
the table, and the quotient will show the number of 
days from the first of January, when the planet will have 
the given longitude. 

EXAMPLES. 

1. On what day of the year 1823, did Venus have 220 
degrees of heliocentric longitude ? Answer— Fourth of 
July. 

2. On what days in the year 1825, did each of the plan- 
ets enter Virgo ? 

problem x. 

To find whether Venus or Mercury will cross the 
Sun's disk in any given year. 

Rule. — Find by Problem 9th when Venus will pass 
her node. Find the earth's heliocentric longitude for 
that day, and if it equals the longitude of Venus' node, 
there will be a transit of Venus, and in no other case. 
The same may be said of the planet Mercury. 

examples. 
Were there a transit of Venus in the year 1824, or not? 
The longitude of the ascending node of Venus, is 75 



SEC. XI.] ASTRONOMICAL PROBLEMS. 103 

degrees and 8 minutes, which she passed on the 26tb 
of June. The earth's longitude on that day, was 274 
degrees and 44 minutes. The longitude of the descend- 
ing node of Venus, was 258 degrees and 8 minutes, 
which she passed on the 5th of March. The earth's 
longitude on that day, was one hundred and sixty -four 
degrees and fifty-five minutes, consequently there was no 
transit of Venus in 1824. 



PROBLEM XI. 

To find when any two given planets shall have a 
given heliocentric aspect, taking their longitudes as 
stated in the Table for 1823. 

Rule. — Add the degrees in the aspect given to the 
heliocentric longitude of either given planet. Find the 
difference between that sum and the heliocentric longi- 
tude of the other given planet : then say, as the differ- 
ence in the daily motions of the two given planets, is 
to one day, so is the difference in their longitude found 
as above to the answer required. 

EXAMPLES. 

1. At what time in the year 1824, did the earth and 
Venus have a trine aspect ? 

The longitude of the earth for January 1st, for that 
year, was 100 degrees and 6 minutes ; to the earth's lon- 
gitude, add 120 degrees, (the given aspect,) and the sum 
is 220 degrees and 6 minutes. 

The longitude of Venus on the first day of January, 
1824, was 150 degrees and 2 minutes ; the difference 
was 70 degrees and 4 minutes ; then 1,6021 degrees — 
,9856=,6165, difference of daily motion. Then ,6165 : 1 
day : : 70 degrees 4 minutes : one hundred and thirteen 
days, or the 22d of April. 

2. On what day were the earth and Jupiter in con- 
junction in the year 1826? 



104 ASTRONOMICAL PROBLEMS. [SEC. XI. 

3. When in 1835, will the earth and Venus be in con- 
junction ? 

Note.— The preceding PROBLEMS would be correct, if the 
Planets moved in perfect circular orbits, which, however, is not the 
fact: yet they approach so near to circles, that deductions founded 
upon their figures as circles, are sufficiently accurate for ordinary 
calculations. 



SEC. XII.] ON ECLIPSES. 105 



SECTION TWELFTH. 



ON ECLIPSES. 

In the solar system, the Sun is the great fountain of 
light, and every planet and satellite is illuminated by him, 
receive the distribution of his rays, and are irradiated by 
his beams. The rays of light are seen in direct lines, 
and consequently are frequently intercepted by the dark 
and opaque body of the Moon, passing directly between 
the earth and the Sun, and hiding a portion or the whole 
of his disk from the view of those parts of the earth where 
the penumbra, or the shadow of the Moon, happens to 
fall. This is called an Eclipse of the Sun. 

It is only at the time of new Moon that an eclipse of 
this kind can possibly take place, and then only when 
the Sun is within seventeen degrees of either the ascend- 
ing or descending nodes ; for if his distance at the time 
of new Moon be greater than seventeen degrees from 
either node, no part of the Moon's shadow will touch the 
earth, and consequently there will be no eclipse. 

The orbit in which the Moon really moves is different 
from the ecliptic, one half being elevated five and one- 
third degrees above it, and the other half as much de- 
pressed below. The Moon's orbit therefore intersects the 
ecliptic in two points diametrically opposite to each other, 
and these intersections are called the Moon's nodes. — 
The Moon, therefore, can never be in the ecliptic but 
when she is in either of her nodes, which is at least 
twice in every lunation, or course from change to change, 
and sometimes thrice. That node from which the Moon 
begins to ascend northward, or above the ecliptic in 
northern latitudes, is called the ascending node ; and the 



106 ON ECLIPSES. [SEC. XII. 

other the descending node ; because the Moon when she 
passes by it descends below the Ecliptic southward. — 
The Ecliptic is the great circle which the earth describes 
in its annual revolution around the Sun, and is divided 
into twelve equal parts, of 30 degrees each, called signs. 
Six of these, namely Aries, Taurus, Gemini, Cancer, 
Leo, and Virgo, are north ; and the other six, to wit, Li- 
bra, Scorpio, Sagitarius, Capricornus, Aquarius, and 
Pisces, south of the equator.* 

When the earth comes between the Sun and Moon, the 
Moon passes through the earth's shadow, and having no 
light of her own, she suffers a real eclipse, the rays of the 
Sun being intercepted by the earth. This can only hap- 
pen at the time of full Moon, and when the Sun is within 
twelve degrees of the Moon's ascending or descending 
nodes. Should the Sun's distance from the node exceed 
twelve degrees, the shadow of the earth would nowhere 
touch the surface of the Moon, and consequently she 
could not suffer an eclipse. 

When the Sun is eclipsed to us, the inhabitants of the 
Moon on the side next the earth, see her shadow like a 
dark spot travelling over the earth about twice as fast as 
its equatorial parts move, and the same way. 

When the earth passes between the Sun and Moon, 
the Sun appears in every part of the Moon where the 
earth's shadow falls totally eclipsed ; and the duration is 
as long as she remains in the earth's shadow. 

If the earth and Sun were of equal sizes, the shadow 
of the earth would be infinitely extended, and wholly of 
the same breadth, and the planet Mars when in either of 
her nodes, and in opposition to the Sun, (although forty- 
two millions of miles from the earth,) would be eclipsed 
by the shadow. If the earth were larger than the Sun, 
her shadow would be sufficient to eclipse the larger 
planets, Jupiter and Saturn, with all their satellites, when 
they were opposite to him ; but the shadow of the earth 
terminates in a point long before it reaches any of the 

* The Equator is an imaginary circle passing round the earth from 
east to west, dividing it into equal parts, called hemispheres. 



SEC. XII.] ON ECLIPSES. 107 

primary planets. [Plate 6th, fig. 8th. S the Sun, AE 
the earth, ABE earth's shadow terminating at B.] It is 
therefore evident, that the earth is much less than the 
Sun, or its shadow could not end in a point at so short a 
distance. 

If the Sun and Moon were of equal sizes, she would 
cast a shadow on the earth's surface of more than two 
thousand miles in breadth, even if it fell directly against 
its centre. But the shadow of the Moon is seldom more 
than one hundred and fifty miles in breadth at the earth, 
unless in total eclipses of the Sun, her shadow strikes on 
the earth in a very oblique direction. 

In annular eclipses, the Moon's shadow terminates in 
a point at some distance before it reaches the earth ; and 
consequently the Moon is much less than the Sun. If 
the Moon were actually thrice its present size, it would 
still in many instances be totally eclipsed. A sufficient 
proof of this is given by her long continuance in the 
earth's shadow during any of her total eclipses. There- 
fore the diameter of the earth is more than three times the 
diameter of the Moon. 

Though all opaque bodies on which the Sun shines, 
have their shadows ; yet such is the magnitude of the 
Sun, and the distances of the planets, that the primaries 
can never eclipse each other. A primary can only 
eclipse its secondary, or be eclipsed by it, and never by 
those except when they are in opposition or conjunction 
with the Sun, as before stated. The primary planets are 
very seldom in such positions, but the Sun and Moon are 
in every month. 

If the Moon's orbit were coincident with the plane of 
the ecliptic, in which the earth wheels its stated courses, 
the Moon's shadow would fall on the earth at every 
change, and the Sun be eclipsed to every part of the 
earth where the penumbra happened to fall. In the same 
manner the Moon would have to travel through the mid- 
dle of the earth's shadow, and be totally eclipsed at every 
full. The duration of total darkness in every instance, 
exceeding an hour and a half. 



108 ON ECLIPSES. [SEC. XII, 

A question like the following naturally arises : — Why 
is it that the Sun is not eclipsed at every change, if the 
Moon actually passes between the Sun and the earth ? 
And why is not the Moon eclipsed at every full, if the 
earth passes between the Sun and Moon in every month? 

One half of the Moon's orbit is elevated 5 degrees and 
twenty minutes above the ecliptic, and the other half is 
as much depressed below it ; and, as before has been ob- 
served, the Moon's orbit intersects the ecliptic in two op- 
posite points, called the Moon's Nodes. 

When these points are in a right line with the centre 
of the Sun at new or full Moon, the Sun, Moon, and 
earth, are all in a right line ; and if the Moon be then 
new, her shadow falls upon the earth ; but if she be full, 
the earth's shadow falls upon her. When the Sun and 
Moon are more than 17 degrees from either of the nodes 
at the time of conjunction, the Moon is generally too 
high or too low in her orbit to cast any part of her sha- 
dow on the surface of the earth. And when the Sun is 
more than 12 degrees from either of the nodes at the time 
of full Moon, the Moon is generally either too high or too 
low to pass through any part of the earth's shadow ; 
therefore in both these cases there can be no eclipse. 

This, however, admits of some variation ; for in apo- 
geal eclipses the solar limit is only sixteen degrees and 
thirty minutes, and in perigeal it is eighteen degrees and 
twenty minutes. When the full Moon is in her apogee* 
she will be eclipsed if she be within ten degrees and 
thirty minutes of the node ; and when in her perigee, if 
within twelve degrees and two minutes. 

The Moon's orbit contains 360 degrees, of which the 
limits of 17 degrees at a mean rate for solar eclipses, and 
twelve for lunar, are only small portions, and the Sun 
generally passes by the nodes only twice in a year, and 
consequently it is impossible that eclipses should happen 
in every month. If the line of the nodes, like the axis of 

* The farthest point of each orbit from the earth's centre is called 
the apogee, and the nearest point is called the perigee. These points 
are directly opposite each other 3 and consequently exactly six signs 
asunder. 



SEC. xi l] on eclipses. 109 

the earth, were carried parallel to itself around the Sun, 
there would be exactly half a year between the conjunc- 
tions of the Sun and nodes. But the nodes shift back- 
ward, or contrary to the earth's annual motion, nineteen 
degrees and twenty minutes every year ; and therefore 
the same node comes round to the Sun nineteen days 
sooner every year than in the one preceding. 173 days, 
therefore, after the ascending node has passed by the Sun, 
the descending node also passes by him. In whatever 
season of the year the luminaries are eclipsed, in 173 days 
after we may expect eclipses about the opposite node. — - 
The nodes shift through all the signs and degrees of the 
ecliptic in 18 years and 225 days, in which time there 
would always be a regular periodical return of eclipses, 
if any number of lunations were completed without a 
fraction. But this never happens ; for if both the Sun 
and Moon should start from a line of conjunction with 
either of the nodes in any point of the ecliptic, the Sun 
would perform 18 annual revolutions and 222 degrees of 
the 19th, and the Moon 230 lunations, and 85 degrees of 
another by the time the node came around to the same 
point of the ecliptic again. 

The Sun would then be 138 degrees from the node, 
and the Moon 85 degrees from the Sun. In 223 mean lu- 
nations after the Sun, Moon, and node, have been in* a 
line of conjunction, they return so nearly to the same 
state again, that the same node which was in conjunction 
with the Sun and Moon at the commencement of these 
lunations, will be within 28 minutes and 12 seconds of a 
degree of a line of conjunction with the Sun and Moon 
again, when the last of these lunations is completed. In 
that time there will be a regular period of eclipses, or rather 
a periodical return of the same eclipse for many ages. In 
this period (which was first discovered by the Chaldeans) 
there are 18 Julian years, 11 days, 7 hours, 43 minutes, 
and 21 seconds, when the 29th day of February in leap 
years, is four times included ; but one day less when in- 
cluded 5 times. Consequently, if to the mean time of 
any eclipse, whether of the Sun or Moon, the above 
10 



110 ON ECLIPSES. [SEC. XII. 

named time be added, you will have the mean time of its 
periodical return. But the falling back of the line of con- 
junctions, or oppositions of the Sun and Moon, namely. 
28 minutes, 12 seconds, with respect to the line of the 
nodes in every period, will wear it out in process of time, 
so that the shadow will not again touch the earth or Moon 
during the space of 12,492 years. Those eclipses of the 
Sun which happen about the ascending node, and begin 
to come in at the north pole of the earth, will continue at 
each periodical return to advance southwardly, until they 
leave the earth at the south pole ; and the contrary with 
those that happen about the descending node, and come 
in at the south pole. From the time that an eclipse of 
the Sun first touches the earth until it completes its peri- 
odical returns, and leaves the same, there will be 77 pe- 
riods, equal to 1388 years. The same eclipse cannot then 
again touch this earth in a less space than 12,492 years, 
as above stated. 

If the motions of the Sun, Moon, and nodes, were the 
same in every part of their orbits, we should need nothing 
more than what has been said to find the exact time of 
all eclipses ; but as this is not the case, we are under the 
necessity of forming tables so constructed, that the mean 
time can be reduced to the true. By the following ex- 
ample, it will be found, that by the true motions of the 
Sun, Moon, and nodes, the eclipse calculated, leaves the 
earth five periods sooner than it would have done by 
mean equable motions. To exemplify this matter more 
fully, I will take the eclipse of the Sun which happened 
in the year 1764, March 21st, Old Style, (or April 1st in 
the new,) according to its mean revolutions, and also 
according to its true equated time. 

The shadow, or penumbra, of the Moon, fell in open 
space at each return without touching the earth ever 
since the creation, until the year of our Lord 1295; then 
on the 14th day of June, at 52 minutes and 59 seconds 
in the morning, Old Style, the Moon's shadow touched 
the earth at the north pole. In each succeeding period 
since that time the Sun has come 28 minutes and 12 



SEC. XII.] ON ECLIPSES. Ill 

seconds nearer the same node, and the Moon's shadow has 
gone more southwardly. In the year 1962, on the 18th of 
July, OldStyle, (or 31st in the new,) at lOhours, 36 minutes, 
21 seconds, in the afternoon, the same eclipse will have 
returned 38 times. The Sun will then be only 24 minutes 
and 45 seconds from the ascending node, and the centre 
of the Moon's shadow will fall a little north of the equator. 
At the end of the next following period, in the year 1980, 
July 29th, Old Style, (or August llth in the new,) at 6 
hours, 19 minutes, and 41 seconds, in the morning, the 
Sun will have receded back three minutes and twenty- 
seven seconds from the ascending node ; the Moon will 
then have a small degree of south latitude, and conse- 
quently cast her shadow a little south of the equator. — 
After this, at every following period, the Sun will be 28 
minutes and 12 seconds farther back from the ascending 
node than at the preceding, and the Moon's shadow will 
continue at each succeeding period to approach nearer 
the south pole, until September 13, Old Style, (or October 
1st in the new,) at 11 hours, 46 minutes, and 22 seconds 
in the morning, in the year 2665, when the eclipse will 
have completed its 77th periodical return, and the sha- 
dow of the Moon leaves the earth at the south pole to re- 
turn no more until the lapse of 12,492 years. But on 
account of the true (or unequable) motions of the Sun, 
Moon, and nodes, the first coming in of this eclipse at the 
north pole of the earth, was on the 24th of June, 1313, at 
3 hours, 57 minutes, and 3 seconds, in the afternoon, and 
it will finally leave the earth at the south pole on the 
18th day of August, (according to New Style,) in the year 
2593, at 10 hours, 25 minutes, and 31 seconds, after- 
noon, at the 72d period. So that the true motions do 
not only alter the true times from the mean ? but they also 
cut off five periods from those of the mean returns of this 
eclipse. 

In any year, the number of eclipses of both luminaries 
cannot be less than two, nor more than seven ; the most 
usual number is four, and it is very rare to have more 
than six. The eclipses of the Sun are more frequent 



112 ON ECLIPSES. [SEC. XII. 

than those of the Moon, because the Sun's ecliptic limits 
are greater than those of the Moon's. (The proportion 
being as 17 is to 12,) yet we have more visible eclipses of 
the Moon than of the Sun ; because eclipses of the Moon 
are seen from all parts of that hemisphere of the earth 
which is next her ; and are equally great to each of those 
parts ; but eclipses of the Sun are only visible to that 
small portion of the hemisphere next him, whereon the 
Moon's shadow happens to fall. 

The Moon's orbit being elliptical, and the earth in one 
of its focusses, she is once at her least distance from the 
earth, and once at her greatest, in every lunation or re- 
volution around the earth. When the Moon changes at 
her least distance from the earth, and so near the node 
that her dark shadow falls on the earth, she appears suf- 
ficiently large to cover the whole disk of the Sun from 
that part on which her shadow falls, and the Sun appears 
totally eclipsed for the space of four minutes. 

But when she changes at her greatest distance from 
the earth, and so near the node that her dark shadow is 
directed towards the earth, her diameter subtends a less 
angle than the Sun's, and therefore cannot hide the whole 
disk from am part of the earth, nor does her shadow 
reach it at thai time ; and to the place over which the 
point f her shadow hangs, the eclipse is annular, and 
the ed^e of the Sun appears like a luminous ring around 
the whole body of the Moon. [Plate 5th, fig. 5th.] — 
When the change happens within 17 degrees of the node, 
and the Moon at her mean distance from the earth, the 
point of her shadow just touches the earth, and the Sun 
is totally eclipsed to that small spot on which the Moon's 
shadow falls ; but the duration of total darkness is not 
of a moment's continuance. The Moon's apparent diam- 
eter, when largest, exceeds the Sun's when least, accord- 
ing to the calculations of modern astronomers, two min- 
utes and five seconds ; the duration of total darkness, 
therefore, may at such time continue four minutes and 
six seconds, casting a shadow on the earth's surface of 
180 miles broad. When the Moon changes exactly in 



SEC. XII.] ON ECLIPSES. 113 

the node, the penumbra is circular on the earth at the 
middle of the general eclipse, because at that time it falls 
perpendicularly on the earth's surface ; but in every other 
moment it falls obliquely, and therefore will be elliptical, 
and the more so, as the time is longer after the middle of 
the general eclipse ; and then much greater portions of 
the earth are involved in the penumbra. 

When the penumbra first touches the earth the gene- 
ral eclipse begins, and it ends when it leaves the earth : 
from the beginning to the end, the Sun appears eclipsed 
in some part of the earth or other. When the penumbra 
touches any place, the eclipse begins at that place, and 
ends when the penumbra leaves it.* When the Moon 
changes exactly in the node the penumbra goes over the 
centre of the earth, as seen from the Moon, and conse- 
quently by describing the longest line possible on the 
earth continues the longest upon it ; namely at a mean 
rate five hours and fifty minutes ; more, if the Moon be 
at her greatest distance from the earth, because she then 
moves slowest, and less if she be at her nearest approach, 
because of her accelerated motion. 

The Moon changes at all hours, and as often in one 
node as in the other, and at all distances from them both, 
at different times as it happens ; the variety of phases of 
eclipses are therefore almost innumerable, even at the 
same places, considering also how variously the same 
places are situated on the enlightened disk of the earth 
with respect to the motion of the penumbra, at the dif- 
ferent hours when eclipses happen. 

When the Moon changes 17 degrees short of her 
descending node, the penumbra just touches the northern 
part of the earth's disk near the north pole, and as seen 
from that place, the Moon appears to touch the Sun, but 
hides no part of him from sight. Had the change been 
as far short of the ascending node, the penumbra would 
have touched the southern part of the disk near the 
south pole. When the Moon changes 12 degrees short 

* Plate 6th, figure 10th, abed represent the Moon's penumbra j 
the arch b d its extent on the earth. 
10* 



114 ON ECLIPSES. [SEC. XII, 

of the descending node ; more than a third part of the 
penumbra, falls on the northern parts of the earth at the 
middle of the general eclipse. Had she changed as far 
past the same node, as much of the other side of the 
penumbra would have fallen on the southern parts of the 
earth ; all the rest in open space. 

When the Moon changes 6 degrees from the node, 
almost the whole penumbra falls on the earth at the 
middle of the general eclipse. Plate 6th, figure 2d, 
represents the number of digits eclipsed up to 12 on the 
right hand, where the eclipse being at the node, is total 
at the equator. 

The further the Moon changes from either node within 
17 degrees of it, the shorter is the penumbra's continu- 
ance on the earth ; because it goes over a less portion of 
the disk. The nearer the penumbra's centre is to the 
equator at the middle of the general eclipse, the longer 
is its duration at places where it is central ; because the 
nearer that any place is to the equator, the greater is the 
circle it describes by the earth's motion on its axis, and 
the place moving quick keeps longer in the penumbra, 
whose motion is the same way with that of the place, 
though faster as has been mentioned. That eclipses of 
the Moon can never happen only at the time of full, and 
the reason why she is not eclipsed at every full, has 
already been mentioned. 

The Moon when totally eclipsed, (though a dark 
opaque body, and shines only by reflection,) is not invi- 
sible, if she be above the horizon, and the sky clear ; but 
generally appears of a dusky colour, which some have 
thought to be her native light. But the true cause of 
her being visible, is the scattered beams of the Sun, bent 
into the earth's shadow by going through the atmo- 
sphere, which being more dense near the earth than at 
considerable heights above it, refracts, or bends the rays 
of the Sun more inward the nearer they are passing by 
the earth's surface, than those rays which go through 
higher parts of the atmosphere where it is less dense ; 



SEC. XII.] ON ECLIPSES, 115 

according to its height, until it be so thin, or rare as to 
lose its refractive power. 

When the Moon goes through the centre of the earth's 
shadow, she is directly opposite to the Sun, yet the Moon 
has been often seen totally eclipsed in the horizon, when 
the Sun was also visible in the opposite part of it ; for 
the horizontal refraction being almost 34 minutes of a 
degree, and the diameter of the Sun and Moon being each 
at a mean state but 32 minutes, the refraction causes 
both luminaries to appear above the horizon, when they 
are actually below it. When the Moon is full at 12 
degrees from either node, she just touches the earth's 
shadow, but does not enter into it. When she is full at 
6 degrees from either node, she is totally, but not cen- 
trally immersed in the earth's shadow, and when she 
passes by the node she takes the longest line possible, 
which is the diameter through it, and such an eclipse, 
(being both total and central, see plate 6th, figure 10th,) 
is of the longest duration, namely, three hours, 57 min- 
utes and 6 seconds from the beginning to the end, if the 
Moon be at her greatest distance from the earth ; and 3 
hours, 37 minutes and 26 seconds, if she be at her least 
distance. 

The reason of this difference is, that when the Moon 
is farthest from the earth, her motions are retarded, but 
when nearest to the earth, her motions are accelerated. 

INTERROGATIONS FOR SECTION TWELFTH. 

Are the rays of light proceeding from the Sun, fre- 
quently intercepted ? 

By what are they intercepted 1 

What is understood by the penumbra ? 

What is an eclipse of the Sun ? 

At what stage of the Moon does an eclipse of the Sun 
happen 1 

How near to either of the nodes must the Sun be to 
suffer an eclipse ? 

Does the Moon's orbit differ from the ecliptic ? 

What is the ecliptic 7 



116 ON ECLIPSES. [SEC. XII. 

What are the Moon's nodes ? 

Why cannot the Sun be eclipsed unless he be within 
17 degrees of the node ? 

How often is the Moon in the ecliptic ? 

Which is called the ascending node ? 

Which is called the descending node ? 

What is an eclipse of the Moon? 

At what stage of the Moon does this happen ? 

How near must the Sun be to either of the nodes, so 
that the Moon can suffer an eclipse ? 

What causes it ? 

Should the same distance from either node at the time 
of full Moon exceed twelve degrees, could the shadow 
of the earth touch the surface of the Moon ? 

As the Moon passes between the Sun and the earth at 
every new Moon, why is not the Sun eclipsed at every 
new Moon ? 

Why is not the Moon eclipsed at every full 1 

What is the farthest point of each orbit from the earth's 
centre called ? 

What the nearest point ? 

How many times a year does the Sun generally pass 
by the nodes ? 

In what time do the nodes pass through all the signs 
of the ecliptic? 

How many lunations after the Sun, Moon and nodes 
have been in conjunction, before they return nearly to 
the same state again ? 

What is a periodical return of an eclipse ? 

Are the motions of the Sun, Moon and nodes the same 
m every part of their orbits ? 

How can the mean time of these conjunctions be 
reduced to the true ? 

How many are the greatest number of eclipses that can 
possibly happen in one year ? 

How many the least ? 

What the most usual number ? 

Which is the most frequent, those of the Sun or Moon ? 

What is the reason ? 



SEC. XII.] ON ECLIPSES. 117 

Are there more visible eclipses of the Moon than of 
the Sun ? 

What is the reason ? 

What is a total eclipse of the Sun ? 

How long can the Moon hide the whole face of the 
Sun from our view ? 

In what part of her orbit must the Moon be to cause a 
total eclipse ? 

What is an annular eclipse ? 

How many miles in diameter would the shadow of the 
Moon be on the earth, in an eclipse when total darkness 
continues four minutes ? 

When the Moon changes exactly in the node, what is 
the form of the shadow, and where does it strike the 
earth ? 

When does an eclipse begin ? 

When does it end ? 

When does it begin and end at any particular place ? 

When the Moon changes 17 degrees short of her de- 
scending node, where will her shadow touch the earth ? 

If as far short of her ascending node, where on the 
earth will her shadow fall ? 

Why in total eclipses of the Moon is she not invisible, 
if she be a dark opaque body ? 

Is it possible for the Moon to be visibly eclipsed while 
the Sun is in sight ? 

When the Moon is mil, within six degrees of either 
node, will she be totally eclipsed ? 

When she passes by the node in the earth's shadow, 
how much of the Moon will be eclipsed ? 

What is the longest time that the Moon can suffer an 
eclipse ? 

What the shortest if she be at her least distance ? 

Why is this difference ? 

What is the time of the longest duration of an eclipse 
of the Sun? 

What the shortest, if the eclipse be central 1 



118 ON THE CONSTRUCTION OF TABLES. [SEC. XIII. 



SECTION THIRTEENTH. 



SHOWING THE PRINCIPLES ON WHICH THE FOLLOWING 
ASTRONOMICAL TABLES ARE CONSTRUCTED, AND THE 
METHOD OF CALCULATING THE TIMES OF NEW AND 
FULL MOONS, AND ECLIPSES BY THEM. 

The nearer that any object is to the eye of an observer, 
the greater is the angle under which it appears. — The 
farther from the eye, the less it appears. 

The diameters of the Sun and Moon subtend different 
angles at different times. And at equal intervals of time, 
these angles are once at the greatest, and once at the 
least, in somewhat more than a complete revolution of 
the luminary through the ecliptic from any given fixed 
star, to the same star again. This proves that the Sun 
and Moon are constantly changing their distances from 
the earth, and that they are once at their greatest dis- 
tance, and once at their least, in a little more than a com- 
plete revolution. 

The gradual differences of these angles are not what 
they would be, if the luminaries moved in circular orbits, 
the earth being supposed to be placed at some distance 
from the centre. 

But they agree perfectly with elliptical orbits, suppos- 
ing the lunar focus of each orbit to be at the centre of 
the earth. 

The farthest point of each orbit from the earth's cen- 
tre, is called the apogee ; and the nearest point the peri- 
gee. These points are directly opposite each other. 

Astronomers divide each orbit into 12 equal parts, 
called signs ; and each sign into 30 equal parts called 
degrees ; each degree into 60 equal parts, called minutes ; 
and each minute into 60 equal parts, called seconds . 



SEC. XIII.] ON THE CONSTRUCTION OF TABLES. 119 

The distance, therefore, of the Sun or Moon from any 
point of its orbit, is reckoned in signs, degrees, minutes 
and seconds. The distance here meant, is that through 
which the luminary has moved from any given point, 
(not the space it falls short thereof,) in coming round 
again, be it ever so little. 

The distance of the Sun or Moon from its apogee at 
any given time, is called its mean anomaly, so that in 
the apogee, the anomaly is nothing, in the perigee, it is 
six signs. 

The motions of the Sun and Moon are observed to be 
continually accelerated from the apogee to the perigee ; 
and as gradually retarded from the perigee to the apo- 
gee ; being slowest of all when the mean anomaly is 
nothing, and swiftest when it is six signs. 

When the luminary is in its apogee or perigee, its 
p]ace is the same as it would be if its motions were 
equable in all parts of its orbit. The supposed equable 
motions are called mean, the unequable are justly called 
the true. 

The mean place of the Sun or Moon is always more 
forward than the true, whilst the luminary is moving 
from its apogee to its perigee ; and the true place is 
always more forward than the mean, whilst the luminary 
is moving from its perigee to its apogee. In the former 
case, the anomaly is always less than six signs ; in the 
latter more. 

It has been discovered by a long series of observations, 
that the Sun goes through the ecliptic, from the vernal 
equinox to the same again, in 365 days, 5 hours, 48 
minutes and 54 seconds. And from the first star of 
Aries, to the same star again, in 365 days, 6 hours, 9 
minutes, and 24 seconds, And from his apogee to the 
same again in 365 days, 6 hours, and 14 minutes. The 
first of these is called the solar year ; the second the 
sidereal, and the third the anomalistic year. The solar 
year is 20 minutes and 29 seconds shorter than the 
sidereal year ; and the sidereal year is 4 minutes and 36 
seconds shorter than the anomalistic. Hence it appears, 



120 ON THE CONSTRUCTION OF TABLES. [SEC. XUI. 

that the equinoctial point, or intersection of the ecliptic 
and equator at the beginning of Aries, goes backward, 
with respect to the fixed stars, and that the Sun's apogee 
goes forward. 

The yearly motion of the earth's or Sun's apogee, is 
found to be one minute and six seconds, which being 
subtracted from the Sun's yearly motion, in longitude, 
the remainder is the Sun's mean anomaly. 

It is also observed, that the Moon goes through her 
orbit from any given fixed star to the same again, in 27 
days, 7 hours, 43 minutes, and 4 seconds, at a mean 
rate ; from her apogee to her apogee again in 27 days, 
13 hours, 18 minutes 5 and 43 seconds : and from the 
Sun to the Sun again in 29 days, 12 hours, 44 minutes, 
and 3^er seconds. This confirms the idea that the Moon's 
apogee moves forward in the ecliptic, and that at a much 
greater rate than the Sun's apogee ; since the Moon is 5 
hours, 55 minutes, and 39 seconds longer in revolving 
from her apogee to her apogee again, than from any star 
to the same again. 

The Moon's orbit crosses the ecliptic in two opposite 
points, which are called her nodes, and it is observed 
that she revolves sooner from any node to the same node 
again, than from any star to the same star again, by 2 
hours, 38 minutes and 27 seconds ; which shows that 
her nodes move backward, or contrary to the order of 
signs in the ecliptic. 

To find the Moon's mean motion in a common year 
of 365 days, the proportion is, as the Moon's period, 27 
days, 7 hours, 43 minutes, 5 seconds, is to her whole 
orbit, or 360 degrees, so is a common year of 365 days, 
to 13 revolutions and 4 signs, 9 degrees, 23 minutes, 
5 seconds. The thirteen, revolutions are rejected, and the 
remainder is taken for the Moon's motion in 365 days. 

To calculate the Moon's mean anomaly : 

The Moon's apogee moves once round her whole orbit 
in 8 years, 309 days, 8 hours, and 20 minutes, or, (add- 
ing two days for leap years,) in 3231 days 8 hours and 20 
minutes. Then, as 3231d. 8h. 20m. is to the whole 



SEC. XIII.] ON THE CONSTRUCTION OF TABLES. 121 

circle, or 360 degrees, so is a common year of 365 days, 
to the motion of the Moon's apogee in one year = 40 de- 
grees, 39 minutes, and 50 seconds. 

From the Moon's mean motion in longitude, during 
one year, s. d. m. s. 

4 9 23 5 subtract 
the motion of her apogee, 1 10 39 50 for the 



same time, and there remains, 2, 28, 43, 15, the 

Moon's mean anomaly in one year. 

To find the mean motion of the Moon's node : 

The Moon's node moves backward round her whole 
orbit in 18 years, 224 days, 5 hours ; therefore for its mo- 
tion in 365 days. 

As 18 years, 224 days, 5 hours, is to the whole circle, 
or 360 degrees, so is the year of 365 days, to the motion 
of the Moon's node in 365 days =^19 degrees, 19 minutes, 
and 43 seconds. 

To find the mean motion of the Moon from the Sun. 

The Moon's mean motion in a common year of 365 
days, is 4 signs, 9 degrees, 23 minutes, and 5 seconds, 
over and above 13 revolutions, and the Sun's apparent 
mean motion in the same time, is 11 signs, 29 degrees, 45 
minutes, and 40 seconds. Then from the Moon's mean 
motion for one year, subtract the mean motion of the Sun 
for the same time, and the remainder will be the mean 
motion of the Moon from the Sun in one year =* 4 signs, 

9 degrees, 37 minutes, and 25 seconds. 

The time in which the Moon revolves from the Sun 
to the Sun again, (or from change to change,) is called 
a lunation, which would always consist of 29 days, 12 
hours, 44 minutes, 3 seconds, 2 thirds and 58 fourths, if 
the motions of the Sun and Moon were always equable. 
Hence 12 mean lunations contain 354 days, 8 hours, 48 
minutes, 36 seconds, 35 thirds, and 40 fourths ; which is 

10 days, 21 hours, 11 minutes, 23 seconds, 24 thirds, and 
20 fourths, less than the length of a common Julian year, 
consisting of 365 days and 6 hours ; and 13 mean luna- 
tions contain 383 days, 21 hours, 32 minutes, 39 seconds. 

11 



122 ON THE CONSTRUCTION OF TABLES. [SEC. XIII. 

38 thirds, and 38 fourths, which exceeds the length of a 
common Julian year by 18 days, 15 hours, 32 minutes, 

39 seconds, 38 thirds, and 38 fourths. 

The mean time of new Moon being found for any given 
year and month, as, suppose for March, 1700, Old Style ; 
if this new Moon happens later than the 11th of March, 
then 12 mean lunations added to the time of this mean 
new Moon, will give the time of the mean new Moon in 
March, 1701, after having thrown off 365 days. But, 
when the mean new Moon happens before the 11th of 
March, we must add 13 mean lunations, to have the 
mean time of mean new Moon in March following, 
always taking care to subtract 365 days in common 
years, and in leap years, 366, from the sum of this ad- 
dition. 

Thus in the year 1700, Old Style, the time of mean 
new Moon in March, was the 8th day, at 16 hours, 11 
minutes, and 25 seconds past four, in the morning of the 
9th day, (according to common reckoning.) To this we 
must add 13 mean lunations, from which subtract 365 
days, because the year 1701 is a common year, and there 
will remain 27 days, 13 hours, 44 minutes, 4 seconds, 38 
thirds, and 38 fourths, for the time of mean new Moon 
in March, in the year 1701. 

By carrying on this addition and subtraction until the 
year 1703, we find the time of new Moon in March that 
year to be on the 6th day, at 7 hours, 21 minutes, 17 
seconds, 49 thirds, and 46 fourths, past noon ; to which 
add 13 mean lunations, and subtract 366 days, (the 
year 1704 being leap year,) and there will remain 24 
days, 4 hours, 53 minutes, 57 seconds, 28 thirds, and 20 
fourths, for the time of mean new Moon in March, 1704. 
In this manner was the first of the following tables con- 
structed to seconds, thirds and fourths, and then written 
to the nearest seconds. 

The reason why we chose to begin the year with 
March, was to avoid the inconvenience of adding a day 
to the tabular time in leap years, after the month of Feb- 
ruary, or subtracting a day therefrom, in January, or 



SEC XIII.] ON THE CONSTRUCTION OF TABLES. 123 

February, in those years ; to which all tables of this 
kind are subject, (which begin the year with January,) 
in calculating the times of new or full Moon. 

The mean anomalies of the Sun and Moon, and the 
Sun's mean distance from the ascending node of the 
Moon's orbit, are set down in table 3d, from 1 to 13 luna- 
tions. 

These numbers for 13 lunations being added to the 
radical anomalies of the Sun and Moon ; and to the 
Sun's mean distance from the ascending node, at the 
time of mean new Moon in March, 1700, (table first,) 
will give their mean anomalies, and the Sun's mean dis- 
tance from the node, at the time of mean new Moon in 
March, 1701 ; and twelve mean lunations more with 
their mean anomalies, &c. added, will give them for the 
time of mean new Moon in March, 1702. 

And thus proceed to continue the table as far as you 
please, always throwing off 12 signs when their sum 
exceeds that number, and setting down the remainder as 
the proper quantity. 

If the numbers belonging to 1700, (in table first y ) be 
subtracted from those of 1800, we shall have their whole 
differences in 100 complete Julian years ; which, accord- 
ingly, we find to be 4 days, 8 hours, 10 minutes, 52 
seconds, 15 thirds, and 40 fourths, with respect to the 
time of new Moon. These being added together 60 times, 
(taking care to throw off a whole lunation when the days 
exceed twenty-nine and a half,) make up 60 centuries, or 
6,000 years, as in table 2d, which was carried on to 
seconds, thirds, and fourths, and then written to the 
nearest seconds. In the same manner were the respec- 
tive anomalies and the Sun's distance from the node 
found for these centurial years, and then (for want of 
room) written to the nearest minutes, which is sufficiently 
exact for whole centuries. 

By means of these two tables, we may readily find the 
time of any new Moon in March, together with the anom- 
alies of the Sun and Moon, and the Sun's mean distance 
from the node at these times within the limits of 6,000 



124 ON THE CONSTRUCTION OF TABLES. [SEC. XIII. 

years, either before or after the 18th century ; and the 
mean time of any new, or full Moon, in any given month 
after March, by means of the third and fourth tables, 
within the same limits, as will be shown in the precepts 
for calculation. 

The table is calculated in conformity to the Old Style, 
for the purpose of calculating eclipses which have made 
their appearances in former ages, and table 16th for those 
after the commencement of the 19th century. 

It would be a very easy matter to calculate the time of 
new or full Moon, if the Sun and Moon moved equably 
in all parts of their orbits. But, we have already shown 
that their places are never the same, as they would be 
by equable motions, except when they are in apogee, or 
perigee, which is when their mean anomalies are either 
nothing or 6 signs. And that their mean places are 
always more forward than their true, whilst the anomaly 
is less than 6 signs ; and their true places more forward 
than their mean, when the anomaly is more. 

Hence it is evident, that whilst the Sun's anomaly is 
less than 6 signs, the Moon will overtake him, or be op- 
posite to him sooner than she would if his motion were 
equable ; and later whilst his anomaly is more than 6 
signs. The greatest difference that can possibly happen 
between the mean and true time of new or full Moon, on 
account of the Sun's motion, is three hours, 48 minutes, 
and 28 seconds ; and that is when the Sun's anomaly is 
three signs one degree, or 8 signs and 29 degrees, sooner 
in the first case, and later in the last. In all other signs 
and degrees of anomaly, the difference is gradually less, 
and vanishes when the anomaly is nothing, or 6 signs. 

The Sun is in his apogee on the 30th of June, and in 
his perigee on the 30th of December, in the present age. 
He is therefore nearer the earth in our winter than in 
summer. The proportional difference of distance deduced 
from the Sun's apparent diameter, at these times, is as 
983 to 1017. 

The Moon's orbit is dilated in winter, and contracted 
in summer ; therefore the lunations are longer in winter 



SEC. XIII.] ON THE CONSTRUCTION OF TABLES. 125 

than in summer. The greatest difference is found to be 
22 minutes and 29 seconds : the lunations are gradually 
increasing in length whilst the Sun is moving from his 
apogee to his perigee, and decreasing in length while he 
is moving from his perigee to his apogee. On this ac- 
count the Moon will be later every time in coming to her 
conjunction with the Sun, or being in opposition to him, 
from December until June ; and sooner from June until 
December, than if her orbit had continued of the same 
size during all the year. 

These differences depend wholly on the Sun's anom- 
aly ; they are therefore put together into one table, and 
called the annual, or first equation of the mean to the true 
syzygy.* [See Table 7th.] This equational difference 
is to be subtracted from the time of the mean syzygy, 
when the Sun's anomaly is less than six signs, and added 
when it is more. At the greatest, it is 4 hours, 10 min- 
utes, and 57 seconds ; viz. 3 hours, 48 minutes, and 28 
seconds, on the Sun's unequal motion ; and 22 minutes 
and 29 seconds on the account of the dilation of the 
Moon's orbit. 

This compound equation would be sufficient, for 
reducing the mean time of new, or full Moon to th 
true, if the Moon's orbit were of a circular form, and her 
motion quite equable in it. But the Moon's orbit is 
more elliptical than the Sun's and her motion in it so 
much the more unequal. The difference is so great, 
that she is sometimes in conjunction with the Sun, or in 
opposition to him sooner, by 9 hours, 47 minutes and 
54 seconds, than she would be, if her motion were equa- 
ble ; and at other times as much later. The former hap- 
pens when her mean anomaly is 9 signs, and 4 degrees ; 
and the latter, when it is 2 signs and 26 degrees. [See 
Table 8th.] At different distances of the Sun from the 
Moon's apogee, the figure of the Moon's orbit becomes 
different. It is longest, or most eccentric, when the Sun 
is in the sign and degree, either with the Moon's apogee, 

* The word syzygy signifies both the conjunction and opposition 
of the Sun and Moon. 

11* 



126 ON THE CONSTRUCTION OF TABLES. [SEC XIII. 

or perigee. Shortest, or least eccentric, when the Sun's 
distance from the Moon's apogee is either three signs, or 
nine signs ; and at a mean state when the distance is 
either one sign and fifteen degrees, four signs and fifteen 
degrees, seven signs and 15 degrees, or ten signs and 
fifteen degrees. When the Moon's orbit is at its greatest 
eccentricity, her apogeal distance from the earth's centre 
is to her perigeal distance therefrom, as 1067 is to 933 ; 
when least eccentric, as 1043 is to 957 ; and when at the 
mean state as 1055 is to 945. 

But the Sun's distance from the Moon's apogee is 
equal to the quantity of the Moon's mean anomaly, at 
the time of new Moon ; and by the addition of 6 signs, 
it becomes equal in quantity to the Moon's mean anomaly, 
at the time of full Moon. A table therefore is constructed 
to answer all the various inequalities, depending on the 
different eccentricities of the Moon's orbit in the syzy- 
gies, and called the second equation of the mean to the 
true syzygy. [See Table 9th.] The Moon's anomaly 
when equated by Table 8th, becomes the proper argu- 
ment for taking out the second equation of time, which 
must be added to the former equated time, when the 
Moon's anomaly is less than six signs, and subtracted 
when the anomaly is more. 

There are several other inequalities in the Moon's 
motion, which sometimes bring on the true syzygy a 
little sooner, and at other times keep it back a little later, 
than it would otherwise be ; but they are so small that 
they may be all omitted except two ; the former, of 
which [see Table 10th,] depends on the difference between 
the anomalies of the Sun and Moon in the syzygies ; 
and the latter [see Table 11th,] depends on the Sun's dis- 
tance from the Moon's nodes at these times. The greatest 
difference arising from the former is 4 minutes and 58 
seconds ; and from the latter 1 minute and 34 seconds. 
Having described the phenomena arising from the ine- 
qualities of the solar and lunar motions, we shall now 
explain the reasons of these inequalities. 

In all calculations and observations relating to the Sun 



SEC. XIII.] ON THE CONSTRUCTION OF TABLES. 127 

and Moon, we have considered the Sun as a moving 
body, and the earth as being at rest ; since all the appear- 
ances are the same, whether it be the Sun, or earth that 
moves. But the truth is that the Sun is at rest, and the 
earth actually moves around him, once in every year, in 
the plane of the ecliptic. Therefore, whatever sign and 
degree of the ecliptic the earth is in at any given time, 
the Sun will then appear to be in the opposite sign and 
degree. 

The nearer any body is to the Sun, the more it is 
attracted by him, and this attraction increases, as the 
square of their distances diminishes, and vice versa. 

The earth's annual orbit is elliptical, and the Sun is 
placed in one of its foci. The remotest point of the 
earth's orbit is called the earth's aphelion, and the nearest 
point of the earth's orbit to the Sun, is called the earth's 
perihelion. When the earth is in its aphelion, the Sun 
appears to be in its apogee ; and when the earth is in its 
perihelion, the Sun appears to be in its perigee. 

As the earth moves from its aphelion to its perihelion, 
it is constantly more and more attracted by the Sun ; 
and this attraction by conspiring in some degree with the 
motion of the earth, must necessarily accelerate its 
motion. 

But, as the earth moves from its perihelion to its aphe- 
lion, it is continually less and less attracted by the Sun ; 
and as their attraction acts then just as much against the 
earth's motion, as it has acted for it in the other half of 
the orbit, it retards the motion in the like degree. 

The faster the earth moves, the faster will the Sun 
appear to move : the slower the earth moves, the slower 
is the Sun's apparent motion. 

The Moon's orbit is also elliptical, and the earth keeps 
constantly in one of its foci. The earth's attraction has 
the same kind of influence on the Moon's motion, that the 
Sun's attraction has on the motion of the earth. There- 
fore, the Moon's motion must be continually accelerated, 
whilst she is passing from her apogee to her perigee ; and 
as gradually retarded in moving from her perigee to her 



128 ON THE CONSTRUCTION OF TABLES. [SEC. XIII. 

apogee. At the time of new Moon, she is nearer to the 
Sun than the earth is at that time, by the whole semi- 
diameter of the Moon's orbit ; which, at a mean state, is 
240,000 miles ; and at the full she is as many miles 
farther from the Sun, than the earth then is. Conse- 
quently, the Sun attracts the Moon more than it attracts 
the earth, in the former case, and less in the latter. The 
difference is greatest, when the earth is nearest the Sun ; 
and least when it is farthest from him. The obvious 
result of this is, that, as the earth is nearest to the Sun in 
winter, and farthest from him in summer ; the Moon's 
orbit must be dilated in winter, and contracted in Sum- 
mer. These are the principal causes of the difference of 
time, that generally happens between the mean and true 
times of conjunction or opposition of the Sun and 
Moon. 

The other two differences, which depend on the dif- 
ferences between the anomalies of the Sun and Moon, 
and upon the Sun's distance from the lunar nodes in the 
syzygies, are occasioned by the different degrees of attrac- 
tion of the Sun and earth upon the Moon, at greater or 
less distances, according to their respective anomalies, 
and to the position of the Moon's nodes, with respect to 
the same. 

If it should ever happen, that the anomalies of both 
the Sun and Moon, were either nothing, or six signs at 
the mean time of new or full Moon ; and the Sun should 
then be in conjunction with either of the Moon's nodes, 
all the above-mentioned equations would then vanish ; 
and the mean and true time of the syzygy, would coin- 
cide ; but if ever this circumstance did happen, we can- 
not expect the like again in many ages afterwards. 
Every 49th lunation returns very nearly to the same 
time of the day as before ; for 49 mean lunations wants 
only 1 minute, 30 seconds, 34 thirds of being equal to 
1477 days. In 2,953,059,085,108 days, there are 100,- 
000,000,000 lunations, exactly, and this is the smallest 
number of natural days, in which any exact number of 
mean lunations are completed. 



SEC. XIH.] ON THE CONSTRUCTION OP TABLES. 129 

The following tables are calculated for the meridian 
of Washington, excepting table first, which is calcu- 
lated for the meridian of London, but they equally serve 
for any other place by adding 4 minutes to the tabular 
time, for every degree that the given place is eastward 
from Washington ; or subtracting 4 minutes for every 
degree that the given place is westward from Wash- 
ington. 

These tables also begin the day at noon, and reckon 
forward to the noon following, for one day. Thus, 
March 31st, at 22 hours, 30 minutes, and 25 seconds of 
tabular time, (in common reckoning,) will be April 1st, 
at 30 minutes, 25 seconds after 10 o'clock in the morning. 

interrogations for section thirteenth. 

Does an object appear at a less angle when far off, 
than when near ? 

Do the Sun and Moon subtend different angles at dif- 
ferent times ? 

Are the angles subtended by the Sun and Moon once 
at the greatest, and once at the least in one revolution ? 

Are these gradual differences the same as they would 
be, if those luminaries moved in circular orbits ?. 

Do they agree perfectly with elliptical orbits ? 

Where must the lower focus of each orbit be placed to 
have them agree ? 

What is meant by the term apogee ? 

What by perigee ? 

Into how many parts do astronomers divide each orbit? 

What is meant by the distance of the Sun or Moon 
from any point of its orbit ? 

What is the distance at any given point, of the Sun or 
Moon from its apogee called ? 

What is the anomaly of the Sun or Moon when in 
apogee ? 

What in perigee ? 

In what part of their orbits are the Sun and Moon con- 
tinually accelerated? 

In what part retarded ? 



130 ON THE CONSTRUCTION OF TABLES. [SEC. XIII. 

What are the mean motions of the Sun and Moon 
called? 

What are the unequable called ? 

In what parts of their orbits are the mean motions for- 
ward of the true ? 

In what part are the true forward of the mean ? 

How many signs is the anomaly in the former case? 

How many in the latter ? 

Does the Moon's apogee move forward in the ecliptic ? 
Does it move faster or slower than the Sun's ? 

Does the Moon revolve sooner from any node to the 
same again, than from any fixed star to the same again? 

If so, what is the difference? 

What is meant by a lunation ? 

Why do astronomers begin the year with March ? 

What does table third contain ? What table first ? 

Why was table first calculated for Old Style ? 

What is the greatest difference between the mean or 
true time of new or full Moon, on account of the Sun's 
motion ? 

Are the lunations longer in winter than in summer ? 

What reason can you advance? What the greatest 
difference ? On what do these differences depend ? - 

What are they called ? Why is this equation not suf- 
ficient to reduce the mean time to the true ? 

Is the Moon's orbit more elliptical than the Sun's ? 

What is meant by the word syzygy ? 

Is the Moon sometimes sooner or later in conjunction 
or opposition with the Sun, than she would be if her mo- 
tions were equable in every part of her orbit ? 

If so, what is the greatest difference ? On what ac- 
count does the Moon's orbit become different ? 

When is it the most eccentric ? When the least ? 

What is equal to the Sun's distance from the Moon's 
apogee ? 

On what does the first of these differences depend ? 
On what the second ? 

What is the remotest point of the earth's orbit called ? 

What is the nearest point to the Sun called? 



sec. xiil] on the construction op tables. 131 

Has the attraction of the earth any influence on the 
motion of the Moon? 
In what case is the motion continually accelerated? 
In what case retarded ? 
Why is the Moon's orbit dilated in winter ? 
Why contracted in summer ? 
For what place are the following tables calculated ? 
By what means do they serve for any other place 1 
At what time do the tables commence the day 7 



132 PRECEPTS TO ASTRONOMICAL TABLES. fsEC. XIV. 



SECTION FOURTEENTH. 



PRECEPTS RELATIVE TO THE FOLLOWING TABLES. 

To calculate the true time of New or Full Moon, and 
Eclipses of the Sun or Moon, by the following Tables. 

If the required new or full Moon be between the years 
1800 and 1900, take out the mean time of new Moon in 
March, for the proposed year, from Table 16th, together 
with the anomalies of the Sun and Moon, and the Sun's 
mean distance from the Moon's ascending node. But if 
the time of full Moon be required in March, add the half 
lunation at the bottom of the page, from Table 3d, with its 
anomalies, &c. to the former numbers, if the new Moon 
falls before the 15th of March ; but if after the 15th of 
March, subtract the half lunation before mentioned, with 
the anomalies, &c. and write down the respective re- 
mainders. 

In these additions and subtractions, observe that 60 
seconds make a minute, 60 minutes make a degree, 30 
degrees make a sign, and 12 signs a circle. 

When the number of signs exceed 12 in addition, re- 
ject 12, and set down the remainder. 

When the number of signs to be subtracted is greater 
than the number you subtract from, add 12 signs to the 
minuend, you will then have a remainder to set down. 

When the required new or full Moon is in any month 
after March, write out as many lunations, with their anom- 
alies, and the Sun's distance from the Moon's ascending 
node, from Table 3d, as the given month is after March, 
setting them regularly below the numbers taken out for 
March ; add all these together, and they will give the 



SEC.XIV.] PRECEPTS TO ASTRONOMICAL TABLES. 133 

mean time of the required new or full Moon, with the 
mean anomalies, and the Sun's mean distance from the 
Moon's ascending node, which are the arguments for 
finding the proper equations. 

The method of calculating an eclipse of the Sun dif- 
fers not from that of calculating the time of new Moon : 
and the directions for finding the time of full Moon 
exactly correspond with an eclipse of the Moon. The 
only guide to find whether an eclipse of the Sun will take 
place at any new Moon, is the Sun's distance from the 
node at that time, namely, (17 degrees,) and an eclipse of 
the Moon at the full, (12 degrees ;) therefore to ascertain 
the time when there will be an eclipse of the Sun, write out 
from Table 3d that number of lunations, with their anom- 
alies, &c, which being added to the numbers from Table 
16th, will make the Sun's distance from the ascending 
node at that time not to exceed or fall short more than 
17 degrees of being either 12 signs or 6 ; in the former 
case there will be an eclipse of the Sun about the ascend- 
ing node, and in the latter about the descending. If past 
the ascending node, and within the above named limits, 
it will be in north latitude ; if short, in south latitude ; if 
short of the descending node, in north latitude ; if past, in 
south latitude. The Sun being in the ascending node 
at 12 signs, which is the same as signs, and in the 
descending at 6, to find at what time there will be an 
eclipse of the Moon, write out the half lunation from the 
bottom of Table 3d, with its anomalies, &c. and place 
it under the numbers taken from Table 16th, under which 
write out from the same Table (3d) that number of luna- 
tions which will make the sum of the Sun's distance from 
either node not to exceed or fall short more than 12 de- 
grees of being either 12 signs or 6 ; in either case there 
will be an eclipse of the Moon. 

With the number of days of the sum, enter Table 4th 
under the given month, and against that number you 
have the day of new or full Moon in the left hand col- 
umn ; which set before the hours, minutes, and seconds 
already found. But, (as it will sometimes happen, if the 

12 



134 PRECEPTS TO ASTRONOMICAL TABLES. [SEC. XIV. 

said number of days fall short of any in the column, under 
the given month, add from Table 3d one lunation, with 
its anomalies, &c. to the aforesaid sum, and you will then 
have a new sum of days, wherewith to enter Table 4th, 
under the given month, where you are sure to find it the 
second time, if the first fails. With the signs and degrees 
of the Sun's anomaly, enter Table 7th, and therewith 
take out the annual, or first equation, for reducing the 
mean to the true syzygy ; taking care to make propor- 
tions in the table for the odd minutes of anomaly, as the 
table gives the equation only for whole degrees. 

Observe in this and every other case of rinding equa- 
tions, that if the signs be at the head of the Table, their 
degrees are at the left hand, and are reckoned down- 
wards. But if the signs be at the foot of the Table, their 
degrees are at the right hand, and are counted upwards ; 
the equation being in the body of the table, under or 
over the signs, in a collateral line with the degrees. The 
terms add, or subtract, at the head or the foot of the ta- 
bles where the signs are found, show whether the equa- 
tion is to be added to the mean time of new or full Moon, 
or subtracted from it. In Table 7th, the equation is to 
be subtracted, if the signs of the Sun's anomaly be found 
at the head of the table ; but it is to be added if the signs 
be at the foot. 

With the signs and degrees of the Sun's anomaly, at 
the mean time of new or full Moon, enter Table 8th, and 
take out the equation of the Moon's mean anomaly, sub- 
tract this equation from her mean anomaly, if the signs 
of the Sun's anomaly be at the head of the table ; but add 
it, if they be at the foot ; the result will be the Moon's 
equated anomaly. 

With the signs and degrees of the Moon's equated 
anomaly, enter Table 9th, and take out the second equa- 
tion, for reducing the mean to the true time of new 
Moon, adding this equation, if the signs of the Moon's 
equated anomaly be at the head of the table ; but sub- 
tracting it if they be at the foot, and the result will be 
the mean time of the new or full Moon, twice equated,— 



SEC. XIV.] PRECEPTS TO ASTRONOMICAL TABLES. 135 

Subtract the Moon's equated anomaly from the Sun's 
mean anomaly, and with the remainder, in signs and de- 
grees, enter Table 10th, and take out the third equation, 
applying it to the former equated time, as the titles add, 
or subtract direct, and the result will be the mean time 
of new, or full Moon thrice equated. With the Sun's 
mean distance from the ascending node, enter Table 
11th, and take out the equation answering to that argu- 
ment; adding it to, or subtracting it from, the thrice 
equated time, as the titles direct ; to which apply the 
equation of natural days, from Table 17th, subtracting it, 
if the clock be faster than the Sun, and adding it, if the 
Sun be faster than the clock ; the result will be the true 
time of new or full Moon, and consequently of an eclipse, 
agreeing with solar time. 

The method of calculating an eclipse for any given 
year, will be shown farther on ; and a few examples com- 
pared with the precepts, will render the whole work plain, 
and easily understood. 

The Tables begin the day at noon, and reckon for- 
ward to the noon following. They are also calculated 
for the latitude and longitude of Washington, excepting 
Table 1st, but serve for any place on the surface of the 
globe, by subtracting four minutes for every degree that 
the place lies west of Washington, from the true solar 
time of conjunction or opposition, and adding four min- 
utes to the true solar time for every degree that the place 
lies eastward of Washington, if Table 16th be used, and 
the same from London, if Table 1st be used ; the result 
will be the true solar time of the new or full Moon, and 
consequently of an eclipse corresponding with the place 
for which the calculations are made. 



136 



ASTRONOMICAL TABLES. 



[SEC. XIV. 

TABLE I.— OLD STYLE. 
The mean time, of new Moon in March, Old Style; with the mean 
anomalies of the Sun and Moon ; and the Sun's mean distance 
from the Moon's ascending node, from the year 1700 to 1800 
inclusive. 



Year 


Mean new 


Sun's mean 


Moon's mean 


Sun's mean dis- 


of 


Moon in March. 


anomaly. 


anomaly. 


tance from the 


Christ. 








node. 




D. H. M. S. 


S. D. M. S. 


S. D. M. S. 


S. D. M. s. 


1700 


8 16 11 25 


8 19 58 48 


1 22 30 37 


6 14 31 7 


1701 


27 13 44 5 


9 8 20 59 


28 7 42 


7 23 14 8 


1702 


16 22 32 41 


8 27 36 51 


11 7 55 47 


8 1 16 55 


1703 


6 7 21 18 


8 16 52 43 


9 17 43 52 


8 9 19 42 


1704 


24 4 53 57 


9 5 14 54 


8 23 20 57 


9 18 2 43 


1705 


13 13 42 34 


8 24 30 47 


7 3 9 2 


9 26 5 30 


1706 


2 22 31 11 


8 13 46 39 


5 12 57 7 


10 4 8 17 


1707 

1708 


21 20 3 50 


9 2 8 50 


4 18 34 13 


11 12 51 18 


10 4 52 27 


8 21 24 43 


2 28 22 18 


11 20 54 5 


1709 


29 2 25 7 


9 9 46 55 


2 3 59 24 


29 37 6 


1710 


18 11 13 43 


8 29 2 47 


13 47 30 


1 7 39 54 


1711 


7 20 2 20 


8 18 18 39 


10 23 35 36 


1 15 42 41 


1712 


25 17 34 59 


9 6 40 51 


9 29 12 42 


2 14 25 43 


1713 


15 2 23 36 


8 25 56 43 


8 9 47 


3 2 28 30 


1714 


4 11 12 13 


8 15 12 35 


6 18 48 52 


3 10 31 17 


1715 


23 8 44 52 


9 3 34 47 


5 24 25 57 


4 19 14 18 


1716 


11 17 33 39 


8 22 50 39 


4 4 14 2 


4 27 17 5 


1717 


1 2 22 5 


8 12 6 32 


2 14 2 8 


5 5 19 52 


1718 


19 23 54 45 


9 28 44 


1 19 39 13 


6 14 2 54 


1719 


9 8 43 22 


8 19 44 37 


11 29 27 18 


6 22 5 41 


1720 


27 6 16 1 


9 8 6 49 


11 5 4 24 


8 48 43 


1721 


16 15 4 38 


8 27 22 41 


9 14 52 29 


8 8 51 29 


1722 


5 23 53 14 


8 16 38 33 


7 24 40 34 


8 16 54 16 


1723 


24 21 25 54 


9 5 45 


7 17 40 


9 25 37 18 


1724 


13 6 14 31 


8 24 16 37 


5 10 5 45 


10 3 40 5 


1725 


2 15 3 7 


8 13 32 29 


3 19 53 50 


10 11 42 52 


1726 


21 12 35 47 


9 1 54 41 


2 25 30 56 


11 20 25 54 


1727 


10 21 24 23 


8 21 10 34 


1 5 19 1 


11 28 28 41 


1728 


28 18 57 3 


9 9 52 46 


10 56 7 


1 7 11 42 


1729 


18 3 45 40 


8 28 48 39 


10 20 44 12 


1 15 14 29 


1730 


7 12 34 16 


8.18 4 31 


9 32 17 


1 23 17 16 


1731 


26 10 6 56 


9 6 26 42 


8 6 9 23 


3 2 17 


1732 


14 18 55 33 


8 25 42 34 


6 15 57 28 


3 10 3 4 


1733 


4 3 44 9 


8 14 58 26 


4 25 45 33 


3 18 5 51 


1734 


23 1 16 49 


9 3 20 38 


4 1 22 39 


4 26 48 53 


1735 


12 10 3 25 


8 22 36 30 


2 11 10 44 


5 4 51 40 


1736 


18 54 2 


8 11 52 22 


20 58 49 


5 12 54 27 


1737 


19 16 26 42 


9 14 34 


11 26 35 55 


6 21 37 29 


1738 


9 1 15 18 


8 19 30 26 


10 6 24 


6 29 40 16 



SEC. XIV.] ASTRONOMICAL TABLES. 



137 



TABLE L— OLD STYLE. 



CONTINUED. 



Year 


Mean new 


Sun's mean 


Moon's mean 


Sun's mean dis- 


of 

Christ. 


Moon in March. 


anomaly. 


anomaly. 


tance from the 




D. H. M. 9. 


S. D. M. S. 


S. D. M. S. 


S. D. M. s. 


1739 


27 22 47 58 


9 7 52 38 


9 12 1 6 


8 8 23 18 


1740 


16 7 36 34 


8 27 8 30 


7 21 49 11 


8 16 26 5 


1741 


5 16 25 11 


8 16 24 27 


6 1 37 16 


8 24 28 52 


1742 


24 13 57 52 


9 4 46 34 


5 7 14 22 


10 3 11 54 


1743 


13 22 46 27 


8 24 2 27 


3 17 2 27 


10 18 14 41 


1744 
1745 


2 7 35 4 


8 13 18 20 


1 26 50 32 


10 19 17 28 


21 5 7 44 


9 1 40 32 


1 2 27 38 


11 28 30 


1746 


10 13 56 20 


8 20 56 24 


11 12 15 43 


6 3 17 


1747 


29 11 29 


9 9 18 36 


10 17 52 49 


1 14 46 19 


1748 


17 20 17 36 


8 28 34 28 


8 27 40 54 


1 22 49 5 


1749 


7 5 6 13 


8 17 50 20 


7 7 28 59 


2 51 52 


1750 


26 2 38 53 


9 6 12 32 


6 13 6 5 


3 9 34 53 


1751 


15 11 27 29 


8 25 28 24 
8 14 4416" 


4 22 54 10 


3 17 37 40 


1752 


3 20 16 6 


3 2 42 15 


3 25 40 27 


1753 


22 17 48 45 


9 3 6 28 


2 8 19 21 


5 4 23 28 


1754 


12 2 37 22 


8 22 22 20 


18 7 26 


5 12 26 15 


1755 


1 11 25 59 


8 11 38 12 


10 27 55 31 


5 20 29 2 


1756 


19 8 58 38 


9 24 


10 3 32 37 


6 29 12 3 


1757 


8 17 47 15 


8 19 16 16 


8 13 20 42 


7 7 14 50 


1758 


27 15 19 54 
17 8 31 


9 7 38 28 
8 26 54 20 


7 18 57 48 


8 15 57 52 


1759 


5 28 45 54 


8 24 39 


1760 


5 8 57 8 


8 16 10 12 


4 8 34 


9 2 3 26 


1761 


24 6 29 47 


9 4 32 24 


3 14 11 6 


10 10 46 27 


1762 


13 15 18 24 


8 23 48 16 


1 23 59 11 


10 18 49 14 


1763 


3 7 1 


8 13 4 8 


3 47 16 


10 26 52 1 


1764 


20 21 39 40 


9 1 26 20 


11 9 24 21 


5 35 2 


1765 


10 6 28 17 
29 4 56 


8 20 42 13 


9 19 12 26 


13 37 49 


1766 


9 9 4 20 


8 24 49 32 


1 22 20 51 


1767 


18 12 49 33 


8 28 20 17 


7 4 37 37 


2 23 38 


1768 


6 21 38 10 


8 17 36 9 


5 14 25 42 


2 8 26 25 


1769 


25 19 10 40 


9 5 58 21 


4 20 2 48 


3 17 9 27 


1770 


15 3 59 26 


8 25 14 13 


2 29 50 53 


3 25 12 14 


1771 


4 12 48 2 


8 14 30 5 


1 9 38 58 


4 3 15 1 


1772 


22 10 20 43 


9 2 52 17 


15 16 4 


5 11 58 3 


1773 


11 19 9 19 


8 22 8 9 
8 11 24 1 


10 25 4 9 
9 4 52 14 


5 20 50 


1774 


1 3 57 55 


5 28 3 37 


1775 


20 1 30 25 


8 29 46 13 


8 10 29 20 


7 6 49 38 


1776 


8 10 19 12 


8 19 2 5 


6 20 17 25 


7 14 49 25 


1777 


27 7 51 51 


9 7 24 17 


5 25 54 31 


8 23 32 26 


1778 


16 16 40 28 


8 26 40 9 


4 5 42 36 


9 1 35 13 


1779 


6 1 29 4 


8 15 56 1 


2 15 30 41 


9 9 38 


1780 


23 23 1 44 


8 4 18 13 


1 21 7 47 


10 18 21 1 


1781 


13 7 50 21 


8 23 34 5 1 


55 52 


10 26 23 48 



12* 



138 



ASTRONOMICAL TABLES. 



SEC. XIV. 



TABLE I.— OLD STYLE. 
CONCLUDED. 



Year 
of 

Christ. 


Mean new 
Moon in March. 

D. H. M. s. 


Sun's mean 
anomaly. 

S. D. M. S, 


Moon's mean 
anomaly. 

s. D. M. s. 


Sun's mean dis- 
tance from the 
node. 
S. D. M. S. 


1782 
1783 

1784 
1785 

1786 
1787 
1788 


2 16 38 57 
21 14 11 37 

9 23 13 
28 20 32 53 
18 5 21 30 

7 14 10 6 
25 11 42 46 

14 20 31 23 

4 5 19 59 

23 2 52 39 

11 11 41 15 


8 12 49 58 

9 1 12 10 

8 20 28 3 

9 8 50 15 
8 28 6 7 

8 17 21 59 

9 5 44 11 


10 10 43 57 
9 16 21 3 
7 26 9 8 
7 1 46 14 
5 11 34 19 
3 21 22 24 
2 26 59 30 


11 4 26 35 
13 9 36 

21 12 23 

1 29 55 25 

2 7 58 12 

2 16 59 

3 24 44 1 


1789 
1790 
1791 
1792 


8 25 3 

8 14 15 55 

9 2 38 7 

8 21 53 59 

9 10 16 11 

8 29 32 3 

8 18 47 55 

9 7 10 7 


1 6 47 35 
11 16 35 40 

10 22 12 46 
9 2 52 

8 7 37 58 
6 17 26 4 
4 27 14 9 
4 2 51 14 

2 12 39 19 
22 27 25 

11 28 4 3 
10 7 52 36 


4 2 46 48 

4 10 49 35 

5 19 32 37 
5 27 35 24 


1793 
1794 
1795 
1796 


30 9 13 55 

19 18 2 32 
9 2 51 8 

27 23 48 


7 6 18 26 
7 14 21 13 
7 22 24 
9 17 1 


1797 
1798 
1799 
1800 


16 9 12 24 

5 18 1 1 

24 15 23 41 

3 22 17 


8 26 25 59 

8 15 41 51 

9 4 4 3 
8 23 19 55 


9 9 9 48 

9 17 12 35 

25 55 37 

11 3 58 24 



SEC. XIV.] ASTRONOMICAL TABLES. 



139 



TABLE II. 

The first new Moon, with the mean anomalies of the Sun and Moon, 
and the Sun's mean distance from the ascending node, next, after 
complete centuries of Julian years. 



Julian 










Years. 


D. H. M. S. 


S. D. M. 


S. D. M. 


S. D. M. 


100 


4 8 10 52 


3 21 


8 15 22 


4 19 27 


200 


8 16 21 44 


6 42 


5 44 


9 8 55 


300 


13 32 37 


10 3 


116 6 


1 28 22 


400 


17 8 43 29 


13 24 


10 1 28 


6 17 49 


500 


21 16 54 21 


16 46 


6 16 50 


11 7 16 


600 


26 1 5 14 


20 7 


3 2 12 


3 26 44 


700 


20 32 3 


11 24 22 


10 21 45 


7 15 31 


800 


5 4 42 55 


11 27 43 


7 7 7 


4 58 


900 


9 12 53 47 


1 4 


3 22 29 


4 24 25 


1000 
2000 


13 21 4 40 


4 25 


7 51 


9 13 53 

6 27 45 


27 18 9 19 


8 50 


15 42 


3000 


12 2 29 56 


11 14 8 


11 27 43 


3 10 58 


4000 


25 23 34 35 


11 18 33 


5 34 


24 50 


5000 


10 7 55 12 


10 23 52 


11 17 36 


9 8 3 


6000 


24 4 59 52 


10 28 17 


11 25 27 


6 21 56 



TABLE III. 

Mean anomalies of the Sun and Moon, and the Sun's mean distance 
from the node, for 13§ mean lunations. 



h 










Sun's mean 


Mean 


Sun's mean 


Moon's mean 


distance from 


§ £> 


lunations. 


anomaly. 


anomaly 




the node. 


w 


D. H. M. s. 


S. D. M. S. 


S. D. M. 


s. 


s. 


D. M. S. 


1 


29 12 44 3 


29 6 19 


25 49 





1 


40 14 


2 


59 1 28 6 


1 28 12 39 


1 21 38 


1 


2 


1 20 28 


2 


88 14 12 9 


2 27 18 58 


2 17 27 


1 


3 


2 42 


4 


118 2 56 12 


3 26 25 17 


3 13 16 


2. 


4 


2 40 56 


5 


147 15 40 15 


4 25 31 37 


4 9 5 


2 


5 


3 21 10 


6 


177 4 24 18 


5 24 37 56 


5 4 54 


3 


6 


4 1 24 


7 


206 17 8 21 


6 23 44 15 


6 43 


3 


7 


4 41 38 


8 


236 5 52 24 


7 22 50 35 


6 26 32 


3 


8 


5 21 52 


9 


265 18 36 27 


8 21 56 54 


7 22 21 


4 


9 


6 2 6 


10 


295 7 20 30 


9 21 3 14 


8 18 10 


4 


10 


6 42 20 


11 


324 20 4 33 


10 20 9 33 


9 13 59 


5 


11 


7 22 34 


12 


354 8 48 36 


11 19 15 52 


10 9 48 


5 





8 2 47 


13 


383 21 32 40 


18 22 12 


11 5 37 


6 


1 


8 41 1 


S'cT 

CO p 


14 18 22 2 


14 33 10 


6 12 54 30 


15 20 7 



Id 3 The half lunation above, is used in finding the mean time of full 
Moon, and likewise in calculating her eclipses. 



140 ASTRONOMICAL TABLES. [SEC. XIV. 

TABLE IV. 

The days of the Year, reckoned from the beginning of 
March. 



£ 



11 n 



62 93 

63 94 



42 72 

43 73 



99 
100 
101 
102 



103 
104 
105 
106 
107 

108 
109 
110 
111 
112 

113 
114 
115 
116 
117 

118 
119 
120 
121 
122 






123 


154 


124 


155 


125 


156 


126 


157 


127 


158 


128 


159 


129 


160 


130 


161 


131 


162 


132 


163 


133 


164 


134 


165 


135 


166 


136 


167 


137 


168 


138 


169 


139 


170 


140 


171 


141 


172 


142 


173 


143 


174 


144 


175 


145 


176 


146 


177 


147 


178 


148 


179 


149 


180 


150 


181 


151 


182 


152 


183 


153 


184 



GQ 

a 

3 
w 

a 


O 
o 

1 


9 

3 

CB 


f o 

a 
o 

CB 

3 

cr 
ft> 


«*3 


2 

CD 

a 


185 


215 


246 


276 


307 


338 


186 


216 


247 


277 


308 


339 


187 


217 


248 


278 


309 


340 


188 


218 


249 


279 


310 


341 


189 


219 


250 


280 


311 


342 


190 


220 


251 


281 


312 


343 


191 


221 


252 


282 


313 


344 


192 


222 


253 


283 


314 


345 


193 


223 


254 


284 


315 


346 


194 


224 


255 


285 


316 


347 


195 


225 


256 


286 


317 


348 


196 


226 


257 


287 


318 


349 


197 


227 


258 


288 


319 


350 


198 


228 


259 


289 


320 


351 


199 


229 


260 


290 


321 


352 


200 


230 


261 


291 


322 


353 


201 


231 


262 


292 


323 


354 


202 


232 


263 


293 


324 


355 


203 


233 


264 


294 


325 


356 


204 


234 


265 


295 


326 


357 


205 


235 


266 


296 


327 


358 


206 


236 


267 


297 


328 


359 


207 


237 


268 


298 


329 


360 


208 


238 


269 


299 


330 


361 


209 


239 


270 


300 


331 


362 


210 


240 


271 


301 


332 


363 


211 


241 


272 


302 


333 


364 


212 


242 


273 


303 


334 


365 


213 


243 


274 


304 


335 




214 


244 


275 


305 


336 






245 




306 


337 





SEC. XIV.] ASTRONOMICAL TABLES. 



141 



TABLE V. 

THE SUN'S DECLINATION. 

argument. — The Sun's true place. 



De- 


Signs. 


Signs. 


Signs. 


De- 


grees. 


ON. 


1 N. 


2N. 


grees. 




6S. 


7S. 


8S. 




D. M. 


D. M. 


D. M. 








11 30 


20 11 


30 


1 


24 


11 51 


20 24 


29 


2 


48 


12 11 


20 36 


28 


3 


1 12 


12 32 


20 48 


27 


4 


1 36 


12 53 


20 59 


26 


5 


1 59 


13 13 


21 10 


25 


6 


2 23 


L3 33 


21 21 


24 


l 


2 47 


13 53 


21 31 


23 


8 


3 11 


14 12 


21 41 


22 


9 


3 34 


14 31 


21 50 


21 


10 


3 58 


14 50 


21 59 


20 


11 


4 22 


15 9 


22 8 


19 


12 


4 45 


15 28 


22 16 


18 


13 


5 9 


15 46 


22 24 


17 


14 


5 32 


16 4 


22 31 


16 


15 


5 55 


16 22 


22 38 


L5 


16 


6 18 


16 39 


22 45 


14 


17 


6 41 


16 57 


22 51 


13 


18 


7 4 


17 14 


22 56 


12 


19 


7 27 


17 30 


23 2 


11 


20 


7 50 


17 46 


23 6 


10 


21 


8 13 


18 2 


23 11 


9 


22 


8 35 


18 18 


23 14 


8 


23 


9 57 


18 33 


23 18 


7 


24 


9 20 


18 48 


23 21 


6 


25 


9 42 


19 3 


22 23 


5 


26 


10 4 


19 17 


23 25 


4 


27 


10 25 


19 31 


23 27 


3 


28 


10 47 


19 45 


23 28 


2 


29 


11 8 


19 58 


23 29 


1 


30 


11 30 


20 11 


23 29 





De- 


Signs. 


Signs. 


Signs. 


De- 


grees. 


11 S. 


10 S. 1 


9S. ! 


grees. 




5N. 


4N. 1 


3N. 





142 



ASTRONOMICAL TABLES. 



!C. XIV. 



TABLE VI. 

EQUATION OP THE SUN's CENTRE, OR THE DIFFERENCE 
BETWEEN HIS MEAN AND TRUE PLACE. 



abgumbnt — Sun's mean anomaly. 



to 

°s 

to 








Subtract. 








1 


Signs. 


1 Sign. 


2 Signs. 


[ 3 Signs. 


4 Signs. 


5 Signs. 


09 




D. M. 


s. 


D. M. 


s. 


D. M. 


s. 


D. M. 


s. 


D. M. 


s. 


D. M. 


s. 


30 





01) 


56 


47 


1 39 


6 


i 55 


3? 


1 41 


12 


58 


30 


1 


1 


59 


58 


30 


1 40 


7 


1 55 


39 


1 40 


12 


57 


7 


29 


2 


3 


57 


1 


12 


1 41 


6 


1 55 


38 


1 39 


10 


55 


19 


28 


3 


5 


50 


1 1 


53 


1 42 


3 


1 55 


36 


1 38 


6 


63 


30 


27 


4 


7 


54 


1 3 


35 


1 42 


59 


1 55 


31 


1 37 





51 


40 


26 


5 
6 


9 


52 


1 5 


12 


1 43 


52 


1 55 


24 


1 35 


52 


49 


49 


25 
24 


11 


50 


1 6 


5'! 


1 44 


44 


1 55 


15 


1 34 


43 


47 


57 


7 


13 


48 


1 8 


2? 


1 45 


34 


1 55 


3 


1 33 


32 


46 


05 


23 


8 


15 


46 


1 10 


2 


1 46 


24 


1 54 


50 


1 32 


19 


44 


11 


22 


9 


17 


43 


1 11 


36 


1 47 


8 


1 54 


35 


1 31 


4 


42 


16 


21 


10 
11 


19 


40 


1 13 


9 


1 47 


53 


1 54 


17 


1 29 


47 


40 


21 


20 
19 


21 


37 


1 14 


41 


1 48 


35 


1 53 


57 


1 28 


29 


38 


25 


12 


23 


33 


1 16 


11 


1 49 


15 


1 53 


36 


1 27 


9 


36 


28 


18 


13 


25 


29 


1 17 


40 


1 49 


54 


1 53 


12 


1 25 


48 


34 


30 


17 


14 


27 


25 


1 19 


8 


1 50 


30 


1 52 


46 


1 24 


25 


32 


32 


16 


15 
16 


29 


20 


1 20 


34 


1 51 


5 


1 52 


18 


1 23 





30 


33 


15 
14 


31 


15 


1 21 


59 


1 51 


37 


1 51 


48 


1 21 


34 


28 


33 


17 


33 


9 


1 23 


22 


1 52 


8 


1 51 


15 


1 20 


6 


26 


33 


13 


18 


35 


2 


1 24 


44 


1 52 


38 


1 50 


41 


1 18 


36 


24 


33 


12 


19 


36 


55 


1 26 


5 


1 53 


3 


1 50 


5 


1 17 


5 


22 


32 


11 


20 
21 


38 


47 


1 27 


24 


1 53 


27 


1 49 


26 


1 15 


33 


20 


30 


10 
9 


40 


39 


1 28 


41 


1 53 


50 


1 48 


4C-, 


1 13 


59 


18 


28 


22 


42 


30 


1 29 


57 


1 54 


10 


1 48 


3 


1 12 


24 


16 


26 


8 


23 


44 


20 


1 31 


11 


1 54 


28 


1 47 


19 


1 10 


47 


14 


24 


7 


24 


46 


9 


1 32 


25 


1 54 


44 


1 46 


32 


1 9 


9 


12 


21 


6 


25 
26 


47 


57 


1 33 


35 


1 54 


58 


1 45 


44 


1 7 


29 


10 


18 


5 

4 


49 


45 


1 34 


45 


1 55 


10 


1 44 


53 


1 5 


49 


8 


14 


27 


51 


32 


1 35 


53 


1 55 


20 


1 44 


1 


1 4 


7 


6 


11 


3 


28 


53 


18 


1 36 


59 


1 55 


28 


1 43 


7 


I 2 


24 


4 


7 


2 


29 


55 


3 


1 38 


3 


1 55 


34 


1 42 


10 





39 


2 


4 


1 


30 

i? 

! 


56 


47 


1 39 


6 


1 55 


37 


1 41 


12 


58 


53 









I 

TO 

2 
2 


11 Signs. 


10 Signs. 


9 Signs. . 


8 Signs. 


7 Signs. 


6 Signs. 












A. 


Id. 













SEC. XIV.] 



ASTRONOMICAL TABLES. 



143 



TABLE VII. 

THE ANNUAL, OR FIRST EQUATION OF THE MEAN TO 
THE TRUE SYZYGY. 



argument — Sun's mean anomaly. 



n 












Subtract. 










O 


a 

1 
























1 





Signs. 


] 


Sign. 


2 


Signs. 


3 Signs. 


4 


Sig 


ns. 


5 Signs. 


05 




H. 


M. 


s. 


H. 


H. 


3. 


H, 


M. 


s. 


H. M. 


s. 


H. 


M. 


s. 


H. M. 


8. 


30 











2 


3 


12 


3 


35 





4 10 


53 


3 


39 


30 


2 7 


45 


1 





4 


18 


2 


6 


55 


3 


37 


10 


4 10 


57 


3 


37 


19 


2 3 


55 


29 


2 





8 


35 


2 


10 


36 


3 


39 


18 


4 10 


55 


3 


35 


6 


2 


1 


28 


3 





12 


51 


2 


14 


14 


3 


41 


23 


4 10 


49 


3 


32 


50 


1 56 


5 


27 


4 





17 


8 


2 


17 


52 


3 


43 


26 


4 10 


39 


3 


30 


30 


1 52 


6 


26 


5 
fi 





21 


24 


2 


21 


27 


3 


45 


25 


4 10 


24 


3 


28 


5 


1 48 


4 


25 

24 





25 


39 


2 


25 


9 


3 


47 


19 


4 10 


4 


3 


25 


35 


1 41 


1 


7 





28 


55 


2 


28 


29 


3 


49 


7 


4 9 


39 


3 


23 





1 39 


56 


23 


« 





34 


11 


2 


31 


57 


3 


50 


50 


4 9 


10 


3 


20 


20 


1 35 


49 


22 


9 





38 


26 


2 


35 


22 


3 


52 


29 


4 8 


37 


3 


17 


35 


1 31 


41 


21 


10 
11 





42 


39 


2 


38 


44 


3 


54 


4 


4 7 


59 


3 


14 


49 


1 27 


31 


20 





46 


52 


2 


42 


3 


3 


55 


35 


4 7 


16 


3 


11 


59 


1 23 


19 


16 


(2 





51 


4 


2 


45 


18 


3 


57 


2 


4 6 


29 


3 


9 


6 


1 19 


5 


18 


1-3 





55 


17 


2 


48 


30 


3 


58 


27 


4 5 


37 


3 


6 


10 


1 14 


49 


17 


14 





59 


27 


2 


51 


40 


3 


59 


49 


4 4 


41 


3 


3 


10 


1 10 


33 


16 


15 
16 


1 


3 


36 


2 


54 


48 


3 


1 


7 


4 3 


40 


3 





7 


1 6 


15 


15 
14 


1 


7 


45 


2 


57 


53 


4 


2 


18 


4 2 


35 


2 


57 





1 1 


56 


17 


1 


11 


53 


3 





54 


4 


3 


23 


4 1 


26 


2 


53 


49 


57 


36 


13 


IS 


1 


16 


8 


3 


3 


51 


4 


4 


22 


4 


12 


2 


50 


36 


53 


15 


12 


19 


1 


20 


6 


3 


6 


45 


4 


5 


18 


3 58 


52 


2 


47 


18 


48 


52 


11 


20 
21 


1 


24 


10 


3 


9 


36 


4 


6 


10 


3 57 


27 


2 


43 


57 


44 


28 


10 

9 


1 


28 


12 


3 


12 


24 


4 


6 


58 


3 55 


59 


2 


40 


33 


40 


2 


22 


I 


32 


12 


3 


15 


9 


4 


7 


41 


3 54 


26 


2 


37 


6 


35 


36 


8 


23 


1 


36 


10 


3 


17 


51 


4 


8 


21 


3 52 


49 


2 


33 


35 


31 


10 


7 


24 


1 


40 


6 


3 


20 


30 


4 


8 


57 


3 51 


9 


2 


30 


2 


26 


44 


6 


25 

26 


1 


44 


1 


3 


23 


5 


4 


9 


29 


3 49 


26 


2 
2 


26 


26 


22 


17 


5 

4 


I 


47 


54 


3 


25 


36 


4 


9 


55 


3 47 


3S 


22 


47 


17 


50 


27 


I 


51 


46 


3 


28 


3 


4 


10 


16 


3 45 


44 


2 


19 


5 


13 


23 


3 


28 


I 


55 


37 


3 


30 


26 


4 


10 


33 


3 43 


45 


2 


15 


20 


8 


56 


2 


29 


1 


59 


26 


3 


32 


45 


4 


10 


45 


3 41 


40 


2 


11 


35 


4 


29 


1 


30 

i 

I 

(A 


2 


3 


12 


3 


35 





4 


10 


53 


3 39 


30 


2 


7 


45 










i 

• era 

3 
a 

! s° 


11 


Signs. 


10 


Signs. 


9 


Signs. 


8 Signs. 


7 


Signs. 


6 Signs. 














Add. 











144 ASTRONOMICAL TABLES. [SEC XIV. 

TABLE VIII. 

EQUATION OF THE MOON'S MEAN ANOMALY. 

argument — Surfs mean anomaly. 



o 


Subtract. 


O 


Signs. 




L Sign. 


2 


Signs. 


3 Signs. 


4 Signs. 


5 Signs. 





D. M. 


3. 


D. 


M. 


s. 


D, 


M. 


s. 


D. M. 


s. 


D. M-. 


s. 


D. M. 


s. 


7. 

30 











46 


45 


1 


11 


32 


1 35 


1 


1 23 


4 


48 


19 


1 


1 


37 





48 


10 


1 


22 


21 


1 35 


2 


1 22 


14 


46 


51 


29 


2 


3 


13 





49 


34 


1 


23 


10 


1 35 


1 


1 21 


24 


45 


23 


23 


3 


4 


52 





50 


53 


1 


23 


57 


1 35 





1 20 


32 


43 


54 


27 


4 


6 


28 





52 


19 


1 


24 


41 


1 34 


57 


1 19 


38 


42 


24 


26 


5 

6 


8 


6 





53 


40 


1 


25 


24 


1 34 


50 


1 18 


42 


40 


53 


25 
24 


9 


42 





55 





1 


26 


6 


1 34 


43 


1 17 


45 


39 


21 


7 


11 


20 





56 


21 


1 


26 


48 


1 34 


33 


1 16 


48 


37 


49 


23 


8 


14 


56 





57 


38 


1 


27 


28 


1 34 


22 


1 15 


47 


36 


15 


22 


9 


33 





58 


56 


1 


28 


6 


1 34 


9 


1 14 


44 


34 


40 


21 


10 
11 


16 


10 


1 





13 


1 


33 


43 


1 33 


53 


1 13 


41 


33 


5 


20 
19 


17 


47 


1 


1 


29 


1 


29 


17 


1 33 


37 


1 12 


37 


31 


31 


12 


19 


23 


1 


2 


43 


1 


29 


51 


1 33 


20 


1 11 


33 


29 


54 


18 


13 


20 


59 


1 


3 


56 


1 


30 


22 


1 33 





1 10 


26 


28 


18 


17 


14 


22 


35 


1 


5 


8 


1 


30 


50 


1 32 


38 


1 9 


17 


26 


40 


16 


15 

16 


24 


10 


1 


6 


18 


1 


31 


19 


1 32 


14 


1 8 


8 


25 


3 


15 
14 


25 


45 


1 


7 


27 


1 


31 


45 


1 31 


50 


1 6 


58 


23 


23' 


17 


27 


19 


1 


8 


36 


1 


32 


12 


1 31 


23 


1 5 


46 


21 


45 


13 


18 


28 


52 


1 


9 


42 


1 


32 


34 


1 30 


55 


1 4 


32 


20 


7 


12 


19 


30 


25 


I 


10 


49 


1 


32 


57 


1 30 


25 


1 3 


19 


18 


28 


11 


20 
21 


31 


57 


1 


11 


54 


1 


33 


17 


1 29 


54 


1 2 


1 


16 


48 


10 

9 


33 


29 


1 


12 


58 


1 


33 


30 


1 29 


20 


1 


45 


15 


8 


22 


35 


2 


I 


14 


1 


1 


33 


52 


1 28 


45 


59 


26 


13 


23 


8 


23 


36 


32 


1 


15 


1 


1 


34 


6 


1 28 


9 


58 


7 


11 


43 


7 


24 


38 


1 


1 


16 





1 


34 


18 


1 27 


30 


56 


45 


10 


7 


6 


25 

26 


39 


29 


1 


16 


59 


1 


34 


30 


1 26 


50 


55 


23 


8 


20 


5 


40 


59 


1 


17 


57 


1 


34 


40 


1 26 


27 


54 


1 


6 


44 


4 


27 


42 


26 


1 


18 


52 


1 


34 


4fi 


1 25 


5 


52 


37 


5 


3 


3 


28 


43 


54 


1 


19 


47 


1 


34 


54 


1 24 


39 


51 


12 


3 


21 


2 


29 


45 


19 


1 


20 


40 


1 


34 


58 


1 23 


52 


49 


45 


1 


40 


1 


30 


47 


45 


1 


21 


32 


1 


35 


1 


1 23 


4 


48 


19 











11 Signs. 


10 


Signs. 


9 


Signs. 


8 Signs. 


7 Signs. 


6 Signs. 


£ 
















Ac 


Id. 












CD 
« 



SEC. XIV.] 



ASTRONOMICAL TABLES. 



its 



TAJBLE IX. 



THE SECOND EQUATION OF THE MEAN TO THE TRUE 
SYZYGY. 

argument — Moon's equated anomaly. 



a> 


Add. 


3 


Signs. 


1 Sign. 


2 Signs. 


3 Signs. 


4 Signs. 


5 Signs. 





H. M. 


s. 


H. M. 


s. 


H. M. 


s. 


H. M. 


s. 


H. M. 


s. 


H. 


M. 


s. 


<y. 








5 12 


48 


8 47 


8 


9 46 


44 


8 8 


59 


4 


34 


33 


30 


1 


10 


56 


5 21 


56 


8 51 


45 


9 45 


3 


8 3 


12 


4 


26 


1 


29 


2 


21 


56 


5 30 


57 


8 56 


10 


9 45 


12 


7 57 


23 


4 


17 


25 


26 


3 


32 


54 


5 39 


51 


9 


25 


9 44 


11 


7 51 


33 


4 


8 


47 


27 


4 


43 


52 


5 43 


37 


9 4 


31 


9 42 


50 


7 45 


46 


4 





7 


26 


5 
6 


54 


50 


5 57 


17 


9 8 


25 


9 41 


36 


7 39 


46 


3 


51 


23 


25 
24 


1 5 


48 


6 5 


51 


9 12 


9 


9 40 


3 


7 33 


36 


3 


42 


32 


7 


1 16 


46 


6 14 


19 


9 15 


43 


9 38 


19 


7 27 


OO 


3 


33 


38 


23 


8 


1 27 


44 


6 22 


41 


9 19 


5 


9 36 


24 


7 21 


2 


3 


24 


42 


22 


9 


1 38 


40 


6 30 


57 


9 22 


14 


9 34 


16 


7 14 


30 


3 


15 


44 


21 


10 
11 


1 49 


33 


6 39 


4 


9 23 


12 


9 32 


1 


7 7 


50 


3 


6 


45 


20 
19 


2 


23 


6 47 





9 27 


54 


9 29 


33 


7 1 


2 


2 


57 


43 


12 


2 11 


10 


6 54 


46 


9 30 


32 


9 26 


54 


6 54 


8 


2 


48 


39 


IS 


13 


2 21 


54 


7 2 


24 


9 32 


58 


9 24 


4 


6 47 


9 


2 


39 


34 


17 


14 


2 32 


34 


7 9 


52 


9 35 


12 


9 21 


3 


6 40 


6 


2 


30 


28 


16 


15 
16 


2 43 


9 


7 17 


9 


9 37 


14 


9 17 


51 


6 32 


56 


2 


21 


19 


15 

14 


2 53 


38 


7 24 


19 


9 39 


8 


9 14 


28 


6 25 


40 


2 


12 


3 


17 


3 4 


3 


7 31 


18 


9 40 


51 


9 10 


54 


6 18 


13 


2 


2 


53 


13 


18 


3 14 


24 


7 38 


9 


9 42 


21 


9 7 


9 


6 10 


49 


1 


53 


36 


12 


19 


3 24 


42 


7 44 


51 


9 43 


42 


9 3 


13 


6 3 


16 


1 


44 


16 


11 


20 
21 


3 34 


58 


7 51 


24 


9 44 


53 


8 59 


6 


5 55 


33 


1 


34 


54 


10 
9 


3 45 


11 


7 57 


45 


9 45 


52 


8 54 


50 


5 47 


54 


1 


25 


31 


22 


3 55 


21 


8 3 


56 


9 46 


38 


8 50 


24 


5 40 


4 


1 


16 


7 


8 


23 


4 5 


26 


8 9 


57 


9 47 


13 


8 45 


48 


5 32 


9 


1 


6 


41 


7 


24 


4 25 


26 


8 15 


46 


9 47 


36 


8 41 


2 


5 24 


9 





57 


13 


6 


25 


4 25 


20 


8 21 


24 


9 47 


49 


8 36 


6 


5 16 


5 





47 


44 


5 

4 


26 


4 35 


6 


8 26 


53 


9 47 


54 


8 31 





5 7 


56 





38 


13 


27 


4 44 


42 


8 32 


11 


9 47 


46 


8 25 


44 


4 59 


42 





28 


41 


3 


28 


4 54 


11 


8 37 


19 


9 47 


33 


8 20 


18 


4 51 


15 





19 


8 


2 


29 


5 3 


33 


8 42 


18 


9 47 


14 


8 14 


33 


4 43 


2 





9 


34 


1 


30 


5 12 


48 


8 47 


8 


9 46 


44 


8 8 


59 


4 34 


33 








3 


, 

S 

TO 

1 


£ 


11 Signs. 


10 Signs. 


9 Signs. 


8Sigi 


is. 


7 Signs. 


6 Signs. 


0(3 






Subtract. 











13 



14# 



ASTRONOMICAL TABLES. [SEC XIV. 



TABLE X. 

THE THIRD EQUATION OF THE MEAN TO THE TRUE SYZYGY. 

Argument.— Sun's mean anomaly— Moon's equated anomaly. 



TO 


Signs. 


Signs. 


Signs. 


b 

en 
ere 

1 P 


Signs, subtract. 


1 Sign, subtract. 


2 Signs, subtract 


3 


6 Signs, add. 


7 Signs, add. 


8 Signs, add. 


ra 


M. S. 


M. S. 


M. S. 








2 22 


4 12 


30 


1 


5 


2 26 


4 15 


29 


2 


10 


2 30 


4 18 


28 


3 


15 


2 34 


4 21 


27 


4 


20 


2 38 


4 24 


26 


5 


25 


2 42 


4 27 


25 


6 


30 


2 46 


4 30 


24 


7 


35 


2 50 


4 32 


23 


8 


40 


2 54 


4 34 


22 


9 


45 


2 58 


4 36 


21 


10 


50 


3 2 


4 38 


20 


11 


55 


3 6 


4 40 


19 


12 


1 


3 10 


4 42 


18 


13 


1 5 


3 14 


4 44 


17 


14 


1 10 


3 18 


4 46 


16 


15 


1 15 


3 22 


4 48 


15 


16 


1 20 


3 26 


4 50 


14 


17 


1 25 


3 30 


4 51 


13 


18 


1 30 


3 34 


4 52 


12 


19 


1 35 


3 38 


4 53 


11 


20 


1 40 


3 42 


4 54 


10 


21 


1 45 


3 45 


4 55 


9 


22 


1 49 


3 48 


4 56 


8 


23 


1 52 


3 51 


4 57 


7 


24 


1 56 


3 54 


4 57 


6 


25 


2 


3 57 


4 57 


5 


26 


2 4 


4 


4 58 


4 


27 


2 9 


4 3 


4 58 


3 


28 


2 13 


4 6 


4 58 


2 


29 


2 18 


4 9 


4 58 


1 


30 


2 22. 


4 12 


4 58 






aq 

2 


Signs. 


! Signs. 


Signs. 


to 
3 

CO 


5 Signs, subtract. 


4 Signs, subtract. 


3 Signs, subtract. 


11 Signs, add. 


10 Signs, add. 


9 Signs, add. 



SEC. XIV.] ASTRONOMICAL TABLES. 



147 



TABLE XL 

THE FOURTH EQUATION OF THE MEAN TO THE TRUE SYZYGY.* 

Argument — The Sun's mean distance from the node. 



Add. 


b 

CD 


Signs. 


1 Sign. 


2 Signs. 


d 

CD 


CfQ 

I 


6 Signs. 


7 Signs. 


8 Signs. 


Cfq 


M. s. 


M. s. 


M. S. 


CD 

5° 








1 22 


1 22 


30 


1 


4 


1 23 


1 21 


29 


2 


7 


1 24 


1 20 


28 


3 


10 


1 25 


1 18 


27 


4 


13 


1 26 


1 16 


26 


5 


16 


1 27 


1 14 


25 


6 


20 


1 28 


1 12 


24 


7 


23 


1 29 


1 10 


23 


8 


26 


1 30 


1 8 


22 


9 


29 


1 31 


1 6 


21 


10 


32 


1 32 


1 3 


20 


11 


35 


1 33 


1 


19 


12 


38 


1 33 


57 


18 


13 


41 


1 34 


54 


17 


14 


44 


1 34 


51 


16 


15 


47 


1 34 


49 


15 


16 


50 


1 34 


45 


14 


17 


52 


1 34 


41 


13 


18 


54 


1 34 


37 


12 


19 


57 


1 33 


34 


11 


20 


1 


1 38 


\ 31 


10 


21 


1 2 


1 32 


28 


9 


22 


1 5 


1 31 


25 


8 


23 


1 8 


1 30 


22 


7 


24 


1 10 


1 29 


19 


6 


25 


1 12 


1 28 


16 


5 


26 


1 14 


1 27 


13 


4 


27 


1 16 


1 26 


10 


3 


28 


1 18 


1 25 


6 


2 


29 


1 20 


1 24 


3 


1 


30 


1 22 


1 22 










5 Signs, 


j 4 Signs. 


3 Signs. 


1 


v 


11 Signs. 


1 10 Signs. 


9 Signs. 


I 


I 




Subtract 




1 



148 



ASTRONOMICAL TABLES. 



[SEC. XIV. 



TABLE XII. 

THE SUN'S MEAN LONGITUDE, MOTION, AND ANOMALY. 

SUN'S MEAN LONGITUDE — SUN'S MEAN ANOMALY- 



Years beginning. 


s. 


D. 


M. 


s. 


s. 


D. 


M, 


s. 


Old Style 


1 


9 


7 


53 


10 


6 


28 


48 






201 


9 


9 


23 


50 


6 


26 


57 






301 


9 


10 


9 


10 


6 


26 


1 






401 


9 


10 


54 


30 


6 


25 


5 






501 


9 


11 


39 


50 


6 


24 


9 






1001 


9 


15 


26 


30 


6 


19 


32 






1101 


9 


16 


11 


50 


6 


18 


36 






1201 


9 


16 


57 


10 


6 


17 


40 






1301 


9 


17 


42 


30 


6 


16 


44 






1401 


9 


18 


27 


50 


6 


15 


49 






1501 


9 


19 


13 


10 


6 


14 


53 






1601 


9 


19 


58 


30 


6 


13 


57 






1701 


9 


20 


43 


50 


6 


13 


1 






1801 


9 


21 


29 


10 


6 


12 


6 




New Style 


1797 


9 


10 


37 


33 


6 


1 


8 


17 




1798 


9 


10 


23 


13 


6 





52 


51 




1799 


9 


10 


8 


54 


6 





37 


26 




1800 


9 


9 


54 


35 


6 





22 


1 




1801 


9 


9 


40 


16 


6 





6 


36 




1802 


9 


9 


25 


56 


5 


29 


51 


10 




1803 


9 


9 


11 


37 


5 


29 


35 


45 




1804 


9 


9 


56 


26 


6 





19 


28 




1805 


9 


9 


42 


6 


6 





4 


2 




1806 


9 


9 


27 


48 


5 


29 


48 


38 




1807 


% 


9 


13 


29 


5 


29 


33 


13 




1808 


9 


9 


58 


17 


6 





16 


48 




1809 


9 


9 


43 


57 


6 





1 


31 




1810 


9 


9 


29 


37 


5 


29 


45 


57 




1811 


9 


9 


15 


17 


5 


29 


30 


32 




1812 


9 


10 





5 


6 





14 


15 




1813 


9 


9 


45 


45 


5 


29 


58 


49 




1814 


9 


9 


31 


25 


5 


29 


43 


26 




1815 


9 


9 


17 


5 


5 


29 


27 


58 ' 




1816 


9 


10 


1 


53 


6 





11 


41 




1817 


9 


9 


47 


33 


5 


29 


56 


15 




1818 


9 


9 


33 


13 


5 


29 


40 


50 




1819 


9 


9 


18 


53 


5 


29 


25 


24 




1820 


9 


10 


3 


41 


6 





9 


7 




1821 


9 


9 


49 


22 


5 


29 


55 


42 



SEC. XIV.] ASTRONOMICAL TABLES. 



149 



TABLE XII.— Continued. 

THE SUN'S MEAN LONGITUDE, MOTION, AND ANOMALY 

sun's mean motion — sun's mean anomaly. 



Years com- 














plete. 


1 s. 


D. 


M. 


s. 


S. D. 


M. 


1 


f n 


29 


45 


40 


11 29 


45 


2 


11 


29 


31 


21 


11 29 


29 


3 


11 


29 


17 


20 


11 29 


14 


4 








1 


50 


11 29 


58 


5 


11 


29 


47 


31 


11 29 


42 


6 


11 


29 


33 


11 


11 29 


27 


7 


11 


29 


18 


52 


11 29 


11 


8 








3 


41 


11 29 


55 


9 


11 


29 


49 


21 


11 29 


40 


10 


11 


29 


35 


2 


11 29 


24 


11 


11 


29 


20 


42 


11 29 


9 


12 








5 


31 


11 29 


53 


13 


11 


29 


51 


12 


11 29 


37 


14 


11 


29 


36 


52 


11 29 


22 


15 
16 


11 




29 



22 

7 


33 


11 29 
11 29 


7 

50 


22 


17 


11 


29 


53 


2 


11 29 


35 


18 


11 


29 


38 


43 


11 29 


20 


19 


11 


29 


24 


23 


11 29 


4 


20 








9 


12 


11 29 


48 


40 








18 


24 


11 29 


37 


60 








27 


36 


11 29 


26 


80 








36 


48 


11 29 


15 


100 








46 





11 29 


4 


200 





1 


32 





11 28 


8 


300 





2 


18 





11 27 


12 


400 





3 


4 





11 26 


16 


500 





3 


50 





11 25 


21 


600 





4 


32 





11 24 


25 


700 





5 


17 


20 


11 23 


29 


800 





6 


2 


40 


11 22 


33 


900 





6 


48 





11 21 


37 


1000 





7 


40 





11 20 


41 


2000 





15 


20 





11 11 


22 


3000 





22 


40 





11 2 


3 


4000 


1 





13 


20 


10 22 


44 


5000 


1 


7 


46 


40 


10 13 


25 


6000 


1 


15 


20 





10 4 


6 



13* 



150 



ASTRONOMICAL TABLES. [SEC. XIV. 



TABLE XII.— Continued. 

THE SUN'S MEAN LONGITUDE, MOTION, AND ANOMALY 
SUN'S MEAN MOTION — SUN's MEAN ANOMALY. 



Months. 


1 s. 


D. 


M. 


s. 


1 S. D. M. 


January, 














8' 


February, 


1 





33 


18 


1 33 


March, 


1 


28 


9 


11 


1 28 9 


April, 
May, 


2 


28 


42 


30 


2 28 42 


3 


28 


16 


40 


3 28 17 


June, 


4 


28 


49 


58 


4 28 50 


July, 


5 


28 


24 


8 


5 28 24 


August, 


6 


28 


57 


26 


6 28 57 


September, 


7 


29 


30 


44 


7 29 30 


October, 


8 


29 


4 


54 


8 29 4 


November, 


9 


29 


38 


12 


9 29 37 


December, 


10 


29 


12 


22 


10 29 11 



SUN S MEAN MOTION AND ANOMALY. — SUN S MEAN MOTION AND ANOMALY. 



Days. 


s. 


D. 


M. 


s. 


| Days. 


s. 


D. 


M. S. 


1 








59 


8 


17 





16 


45 22 


2 





1 


58 


17 


18 





17 


44 30 


3 





2 


57 


25 


19 





18 


43 38 


4 





3 


56 


33 


20 





19 


42 47 


5 





4 


55 


42 


21 





20 


41 55 


6 





5 


54 


50 


22 





21 


41 3 


7 





6 


53 


58 


23 





22 


40 12 


8 





7 


53 


7 


24 





23 


39 20 


9 





8 


52 


15 


25 





24 


38 28 


10 





9 


51 


23 


26 





25 


37 37 


11 





10 


50 


32 


•27 





26 


36 45 


12 





11 


49 


40 


28 





27 


35 53 


13 





12 


48 


48 


29 





28 


35 2 


14 





13 


47 


57 


30 





29 


34 10 


15 





14 


47 


5 


31 





30 


33 18 


16 





15 


46 


13 











SEC. XIV.] ASTRONOMICAL TABLES. 



151 



TABLE XII.— Concluded. 

THE SUN'S MEAN LONGITUDE — MOTION AND ANOMALY, 

sun's mean motion and anomaly. 





| Sun's mean 


Sun's mean 




Sun's mean 


Sun's mean 




motion 


and 


distance from 




motion 


and 


distance from 


H. 


D. 


anomaly. 
M. s. 


D, 


the node. 




anomaly. 




the n< 


)de. 


M. 


s. 


H. 


D. M. 


s. 


D. 


M. 


s. 


M. 


M. 


9. 


3ds 


M, 


S. 


T. 


M. 


M. s. 


T. 


M. 


S. 


T. 


S. 
1 


S. 


3ds 


4ths 


S. 


T. 


F. 


S. 

31 


s. T. 


F. 


s. 

1 


T. 


F. 





2 


28 





2 


36 


1 16 


23 


20 


30 


2 





4 


56 





5 


12 


32 


1 18 


51 


1 


23 


6 


3 





7 


24 





7 


48 


33 


1 21 


19 


1 


25 


42 


4 





9 


51 








23 


34 


1 23 


47 


1 


28 


18 


5 

6 





12 


19 





12 


59 


35 

36 


1 26 


15 


1 


30 


54 





14 


47 





15 


35 


1 28 


42 


1 


33 


29^ 


/ 





17 


15 





18 


11 


37 


1 31 


10 


1 


36 


5 


8 





19 


43 





20 


47 


38 


1 33 


38 


1 


38 


40 


1 9 





22 


11 





23 


23 


39 


1 36 


6 


1 


41 


16 


10 
11 





24 


28 





25 


58 


40 
41 


1 38 


34 


1 


43 


52 





27 


6 





28 


34 


1 41 


2 


1 


46 


28 


12 





29 


34 





31 


10 


42 


1 43 


30 


1 


49 


44 


13 





32 


2 





33 


45 


43 


1 45 


57 


1 


51 


39 


14 





34 


30 





36 


21 


44 


1 48 


25 


1 


54 


15 


15 

16 





36 


58 





38 


57 


45 

46 


1 50 


53 


1 


55 


51 





39 


26 





41 


33 


1 53 


21 


1 


59 


27 


17 





41 


53 





44 


8 


47 


1 55 


49 


2 


2 


3 


18 





44 


21 





46 


44 


48 


1 58 


17 


2 


4 


39 


19 





46 


49 





49 


20 


49 


2 


44 


2 


7 


13 


20 
21 





49 


17 





51 


56 


50 
51 


2 3 


12 


2 


9 


50 





51 


45 





54 


32 


2 5 


40 


2 


12 


25 


22 





54 


13 





57 


8 


52 


2 8 


8 


2 


15 


2 


23 





56 


40 





59 


43 


53 


2 10 


36 


2 


17 


38 


24 





59 


8 


1 


2 


19 


54 


2 13 


4 


2 


20 


14 


25 
26 


1 


1 


36 


1 


4 


55 


55 
56 


2 15 


32 


2 


22 


50 


1 


4 


4 


1 


7 


31 


2 17 


59 


2 


25 


26 


27 


1 


6 


32 


1 


10 


7 


57 


2 20 


27 


2 


28 


2 


28 


1 


9 





1 


12 


43 


58 


2 22 


55 


2 


30 


38 


29 


1 


11 


28 


1 


15 


19 


59 


2 25 


23 


2 


33 


14 


30 


1 


13 


55 


1 


17 


55 1 


60 


2 27 


51 


2 


35 


50 



In leap-year after February, add one day, and one day's motion. 



152 



ASTRONOMICAL TABLES. [SEC. XIV. 

TABLE XIII. 



EQUATION OF THE SUNS MEAN DISTANCE FROM THE 
NODE. 

Argument. — Sun's mean anomaly. 



o 








Subtract. 








«i 


















a 





Signs. 


ISi 


gn- 


2 


Signs. 


3Sig 


us. 


4! 


Signs. 


5 Signs. 


w 


D. 


M. 


D. 


M. 


D. 


Mi, 


D. 


M. 


D. 


M. 


D. 


M. 


CD 











1 


2 


1 


47 


2 


5 


1 


50 


1 


4 


30 


1 





2 


1 


4 


1 


48 


2 


5 


1 


48 


L 


2 


29 


2 





4 


1 


6 


1 


49 


2 


5 


1 


47 


1 





28 


3 





6 


1 


8 


1 


50 


2 


5 


1 


46 





58 


27 


4 





9 


1 


10 


1 


51 


2 


5 


1 


45 





56 


26 


5 

6 





11 


1 


12 
14 


1 


52 


2 


5 


1 


44 





54 


25 
24 





13 


1 


1 


53 


2 


5 


1 


43 





52 


7 





15 


1 


16 


1 


54 


2 


4 


1 


41 





50 


23 


8 





17 


1 


17 


1 


55 


2 


4 


1 


40 





48 


22 


9 





19 


1 


18 


1 


56 


2 


4 


1 


38 





46 


21 


10 
11 





21 


1 


19 


1 


57 


2 


4 


1 


37 






44 


20 
19 





23 


1 


21 


1 


58 


2 


3 


1 


36 


42 


12 





25 


1 


22 


1 


58 


2 


3 


1 


34 





40 


18 


13 





28 


1 


24 


1 


59 


2 


3 


1 


33 





37 


17 


14 





30 


1 


26 


2 





2 


2 


1 


31 





35 


16 


15 
16 





32 


1 


27 


2 





2 


2 


1 


30 





33 


15 
14 





34 


1 


28 


2 


1 


2 


1 


1 


28 





31 


17 





36 


1 


30 


2 


1 


2 


1 


1 


27 





29 


13 


18 





38 


1 


31 


2 


2 


2 





I 


25 





27 


12 


19 


9 


40 


1 


34 


2 


2 


2 





I 


24 





24 


11 


20 
21 





42 


1 


35 


2 




1 


59 


1 


23 





22 


10 

9 





44 


1 


36 


2 


3 


1 


59 


1 


21 





20 


29 





46 


1 


37 


2 


4 


1 


58 


I 


19 





18 


8 


23 





48 


1 


39 


2 


4 


1 


57 


1 


17 





16 


7 


24 





50 


1 


40 


2 


4 


1 


56 


1 


15 





13 


6 


25 





52 


1 


41 


2 


4 


1 


55 


1 


13 





11 


5 

4 





54 


1 


43 


2 


5 


1 


54 


1 


11 





9 


27 





56 


1 


44 


2 


5 


1 


53 


1 


9 





7 


3 


98 





58 


1 


45 


2 


5 


1 


52 


1 


3 





5 


2 


29 


1 





1 


46 


2 


5 


1 


51 


1 


6 





3 


1 


30 


1 


2 


1 


47 


2 


5 


1 


50 


1 


4 










O 


11 


Signs. 


10 s 


gns. 


1 9 Signs. 


8 Signs. 


7 


Signs. 


6 


Signs. 






















Add. 











SEC. XIV.] ASTRONOMICAL TABLES. 



153 



TABLE XIV. 

THE MOON'S LATITUDE IN ECLIPSES. 
Argument — Momi's Equated Distance from the Node. 



Signs — North Ascending, 


6 Signs — South Ascending. 


Degrees. J d. m. s. 


Degrees. 








30 


1 


5 15 


29 


2 


10 30 


28 


3 


15 45 


27 


4 


20 59 


26 


5 


26 13 


25 


6 


31 26 


24 


7 


36 39 


23 


8 


41 51 


22 


9 


47 22 


21 


10 


52 13 


20 


11 


57 23 


19 


12 


1 2 31 


18 


13 


1 7 38 


17 


14 


1 12 44 


16 i 


15 


1 17 49 


15 


16 


1 22 52 


14 


17 


1 27 53 


13 


18 


1 32 52 


12 


19 


1 37 49 


11 


5 Signs — North Descenc 


ling. 


11 Signs — South Descenc 


ling. 



This Table shows the Moon's true latitude a little beyond 
the utmost limits of Eclipses. 



154 



ASTRONOMICAL TABLES. 
TABLE XV. 



{sec. XIV. 



The Moon's horizontal parallax, with the semidiameters, and the true 
horary motions of the Sun and Moon, to every sixth degree of their 
mean anomalies ; the quantities for the intermediate degrees being 
easily proportioned by sight. 



Anomaly of the 
and Moon. 


f 

2 w 

P 3* 
320 

* N 


m 

3 
3^ 
01* 

g 

£" 
B" 

B 


Ct> CD 


| 

3 

TO* 

II 

* P 
>-( 


m 

a 

3 

CO* 

f 

3 


> 
gj 


w 

p 

3 


3 

E 


<? 




3 

o 


5' 

3 


02 

3 - 
3 


S. D. | 


M. S. | 


M. S. 


M. S. 


M. S. 


M. S. 


S. D. 





54 29 


15 50 


14 54 


30 10 


2 23 


12 


6 


54 31 


15 50 


14 55 


30 12 


2 23 


11 24 


12 


54 34 


15 50 


14 56 


30 15 


2 23 


11 18 


18 


54 40 


15 51 


14 57 


30 19 


2 23 


11 12 


24 


54 47 


15 51 


14 58 


30 26 


2 23 


11 6 


1 


54 56 


15 52 


14 59 


30 34 


2 24 


11 


1 6 


55 6 


15 53 


15 1 


30 44 


2 24 


10 24 


1 12 


55 17 


15 54 


15 4 


30 55 


2 24 


10 18 


* 1 18 


55 29 


15 55 


15 8 


31 9 


2 24 


10 12 


1 24 


55 42 


15 56 
15 58 


15 12 


31 23 


2 25 


10 6 


2 


55 56 


15 17 


31 40 


2 25 


10 


2 6 


56 12 


15 59 


15 22 


31 56 


2 26 


9 24 


2 12 


56 29 


16 1 


15 26 


32 17 


2 27 


9 18 


2 18 


56 48 


16 2 


15 30 


32 39 


2 27 


9 12 


2 24 


57 8 


16 4 


15 36 


33 11 


2 28 


9 6 
9 


3 


57 30 


16 6 


15 41 


33 23 


2 28 


3 6 


57 52 


16 8 


15 47 


33 47 


2 29 


8 24 


3 12 


58 12 


16 10 


15 52 


34 11 


2 29 


8 18 


3 18 


58 31 


16 11 


15 58 


34 24 


2 29 


8 12 


3 24 

4 


58 49 


16 13 


16 3 


34 58 


2 30 


8 6 


59 6 


16 14 


16 9 


35 22 


2 30 


8 


4 6 


59 21 


16 15 


16 14 


35 45 


2 31 


7 24 


4 12 


59 35 


16 17 


16 19 


36 


2 31 


7 18 


4 18 


59 48 


16 19 


16 24 


36 20 


2 32 


7 12 


4 24 


60 


16 20 


16 28 


36 40 


2 32 


7 6 


5 


60 11 


16 21 


16 31 


37 


2 32 


7 


5 6 


60 21 


16 21 


16 32 


37 10 


2 33 


6 24 


5 12 


60 30 


16 22 


16 37 


37 19 


2 33 


6 18 


5 18 


60 38 


16 22 


16 38 


37 28 


2 33 


6 12 


5 24 


60 45 


16 23 


16 39 


37 36 


2 33 


6 6 


6 


60 45 


I 16 23 


1 16 39 


37 40 


2 33 


6 



SEC. XIV.] 



ASTRONOMICAL TABLES, 



155 



TABLE XVI. 

Mean new Moon, fyciin March, New Style, from 1800 to 1900 inclusive : 
calculated for the meridian of Washington, 76 degrees and 56 min- 
utes west longitude from London. 



Year 


Mean new 


Sun's mean 


Moon's mean 


Sun's mean dis- 


of 

Christ. 


Moon in March. 


anomaly. 


anomaly. 


tance from the 




D. H. M. S. 


S. D. M. S. 


S. D. M. S. 


S. D. M. S. 


1800 


24 19 14 33 


8 23 19 55 


10 7 52 36 


11 3 58 24 


1801 


14 4 3 20 


8 12 35 47 


8 17 40 41 


11 12 1 10 


1802 


3 12 51 46 


8 1 51 39 


6 27 28 46 


11 20 3 57 


1803 


22 10 24 25 


8 20 13 51 


6 3 5 52 


28 46 58 


1804 


10 19 13 2 


8 9 29 43 


4 12 53 57 


1 6 49 45 


1805 
1806 


4 1 39 


7 28 45 31 


2 22 42 2 


1 14 52 35 


19 1 34 18 


8 17 7 43 


1 28 19 8 


2 23 35 36 


1807 


8 10 22 55 


8 6 23 35 


8 7 13 


3 1 38 23 


1808 


26 7 55 35 


8 24 45 47 


11 13 44 19 


4 10 21 24 


1809 


15 16 44 11 


8 14 1 39 


9 23 32 24 


4 18 24 11 


1810 
1811 


5 1 32 48 


8 3 17 31 


8 3 20 29 


4 26 26 58 


23 23 5 28 


8 21 39 43 


7 8 57 35 


6 5 9 59 


1812 


12 7 54 4 


8 10 55 35 


5 18 45 40 


6 13 12 46 


1813 


1 16 42 41 


8 11 27 


3 28 33 45 


6 21 15 23 


1814 


20 14 15 20 


8 18 33 39 


3 4 10 51 


7 29 58 24 


1815 


9 23 3 57 


8 7 53 31 


1 13 58 58 


8 8 1 11 


1816 


27 20 36 37 


8 26 15 43 


19 36 2 


9 16 44 12 


1817 


17 5 25 13 


8 15 31 35 


10 29 24 7 


9 24 46 59 


1818 


6 14 13 50 


8 4 47 27 


9 9 12 12 


10 2 49 46 


1819 


25 11 46 29 


8 23 9 39 


8 14 49 18 


11 11 32 47 


1820 


13 20 35 6 


8 12 25 31 


6 24 37 23 


11 19 35 34 


1821 


3 5 23 43 


8 1 41 23 


5 4 25 23 


11 27 38 21 


1822 


22 2 56 22 


8 20 3 35 


4 10 2 29 


1 6 22 22 


1823 


11 11 44 59 


8 9 19 27 


2 19 50 34 


1 14 25 9 


1824 


29 9 17 39 


8 27 41 39 


1 25 27 40 


2 23 9 10 


1825 


18 18 6 15 


8 16 57 31 


5 15 45 


3 1 11 57 



156 



ASTRONOMICAL TABLES. 



[SEC. XIV. 



TABLE XVI.— Continued. 



Year Mean new 


Sun's mean 


Moon's mean 


Sun's mean dis- 


of 


Moon in March. 


anomaly. 


anomaly. 


tance from the 


Christ. 








node. 




D. H. M. S. 


S. D. M. S. 


s. D. M. s. 


S. D. M. S. 


1826 


8 2 54 52 


8 6 13 23 


10 15 3 50 


3 9 14 44 


i 1827 


27 27 31 


8 24 35 35 


9 20 40 56 


4 17 58 45 


1828 


15 9 16 8 


8 13 51 27 


8 29 1 


4 26 1 32 


1829 


4 18 4 45 


8 3 7 19 


6 10 17 6 


5 4 4 19 


1830 
1831 


23 15 37 24 


8 21 29 31 


5 15 54 12 


6 12 48 20 


13 26 1 


8 10 45 23 


3 25 42 17 


6 20 51 7 i 


1832 


1 9 14 37 


8 1 15 


2 5 30 22 


6 28 53 54 


1833 


20 6 47 17 


8 18 23 27 


1 11 7 28 


8 7 37 55 


1834 


9 15 35 54 


8 7 39 19 


11 20 55 33 


8 15 40 42 


1835 


28 13 8 33 


8 26 1 31 


10 26 32 39 


9 24 24 43 


1836 


16 21 57 10 


8 15 17 23 


9 6 20 44 


10 2 27 30 


1837 


6 6 45 46 


8 4 33 15 


7 16 8 49 


10 10 30 17 


1838 


25 4 18 26 


8 22 55 27 


6 21 45 55 


11 19 14 18 


1839 


14 13 7 2 


8 12 11 19 


5 1 34 


11 27 17 5 


1840 
1841 


2 21 55 39 


8 1 27 11 


3 11 22 5 


5 19 52 


21 19 28 19 


8 19 49 23 


2 16 59 11 


1 14 3 53 


1842 


11 4 16 55 


8 9 5 15 


26 47 16 


1 22 6 40 


1843 


30 1 49 35 


8 27 27 27 


11 32 24 22 


3 50 41 


1844 


18 10 38 12 


8 16 43 19 


10 12 12 27 


3 8 53 28 


1845 
1846 


7 19 26 48 


8 5 59 11 


8 22 32 


3 16 56 15 


26 16 59 28 


8 24 21 23 


7 27 37 38 


4 25 40 16 


1847 


16 1 48 5 


8 13 37 15 


6 7 25 43 


5 3 43 3 


1848 


4 10 36 41 


8 2 53 7 


4 17 13 48 


5 11 45 50 


1849 


23 8 9 21 


8 21 15 19 


3 22 50 54 


6 20 29 51 


1850 
1851 


12 16 57 57 


8 10 31 11 


2 2 38 59 


6 23 32 38 


2 1 46 33 


7 29 47 3 


12 27 4 


7 6 35 25 


1852 


19 23 19 13 


8 18 9 15 


11 18 4 10 


8 15 19 26 


1853 


9 8 7 49 


8 7 25 7 


9 27 52 15 


8 23 22 13 


1854 


28 5 40 29 


8 25 47 19 


9 3 29 21 


10 2 6 14 


1855 


17 14 29 5 


8 15 3 11 


7 13 17 26 


10 10 9 1 



SEC. XIV. 



ASTRONOMICAL TABLES. 



157 



TABLE XVI.— Continued. 



Year 


Mean new 


Sun's mean 


Moon's mean 


Sun's mean dis- 


of 


Moon in March. 


anomaly. 


anomaly. 


tance from the 


Christ. 








node. 




D. H. M. S. 


S. D. M. S. 


S. D. M. S. 


S. D. M. S. 


1856 


5 23 17 42 


8 4 19 2 


5 23 5 31 


10 18 11 48 


1857 


24 20 50 22 


8 22 41 15 


4 28 42 37 


11 26 55 49 


1858 


14 5 38 58 


8 11 57 7 


3 8 30 42 


4 58 36 


1859 


3 14 27 35 


8 1 12 59 


1 18 18 47 


13 1 23 


1860 


21 12 15 


8 19 35 11 


23 55 53 


1 21 45 24 


1861 


10 20 48 51 


8 8 51 3 


11 3 43 58 


1 29 48 11 


1862 


29 18 21 31 


8 27 13 15 


10 9 21 4 


3 8 32 12 


1863 


19 3 10 7 


8 16 29 7 


8 19 9 9 


3 16 34 59 


1864 


7 11 58 44 


8 5 45 


6 28 57 14 


3 24 37 46 


1865 


26 9 31 24 


8 24 7 12 


6 4 34 20 


5 3 21 47 


1866 


15 18 20 


8 13 23 4 


4 14 22 25 


5 11 24 34 


1867 


5 3 8 37 


8 2 38 56 


2 24 10 30 


5 19 27 21 


1868 


23 41 16 


8 21 1 8 


1 29 47 36 


6 28 11 22 


1869 


12 9 29 53 


8 10 17 


9 35 41 


7 6 14 9 


1870 


1 18 18 30 


7 29 32 52 


10 19 23 46 


7 14 16 56 


1871 


20 15 51 9 


8 17 55 4 


9 25 5 


8 23 57 


1872 


9 39 46 


8 7 10 56 


8 4 48 10 


9 1 3 44 


1873 


27 22 12 26 


8 25 33 8 


7 10 25 16 


10 9 47 45 


1874 


17 7 1 2 


8 14 49 


5 20 13 21 


10 17 50 32 


1875 


6 15 49 39 


8 4 4 52 


4 1 26 


10 25 53 19 


1876 


24 13 22 18 


8 22 27 4 


3 5 38 32 


4 37 20 


1877 


13 22 10 55 


8 11 42 56 


1 15 26 37 


12 40 7 


1878 


3 6 59 32 


8 58 48 


11 25 14 42 


20 42 54 


1879 


22 4 32 11 


8 19 21 


11 51 48 


1 29 26 55 


1880 


10 13 20 48 


8 8 36 52 


9 10 39 53 


2 7 29 42 


1881 


29 10 53 28 


8 26 59 5 


8 16 16 59 


3 16 13 43 


1882 


18 19 42 4 


8 16 14 57 


6 26 5 4 


3 24 16 30 


1883 


8 4 30 41 


8 5 30 49 


5 5 53 9 


4 2 19 17 


1884 


26 2 3 20 


8 23 53 1 


4 11 30 15 


5 11 3 18 


1885 


15 10 51 57 


8 13 8 53 


2 21 18 20 


5 19 6 5 



14 



158 



ASTRONOMICAL TABLES. 



[sec, 



XIV. 



TABLE XVI.— Concluded, 













Year 

of 
Clurist. 


Mean new Moon 
in March. 

D. H. M. S. 


Sun's mean 
anomaly. 

S. D. M. S. 


Moon's mean 
anomaly. 

S. D. M. S. 1 


Sun's mean dis- 
tance from the 

node. 
S. D. M. S. 


1886 
1887 
1888 
1889 
1890 


4 19 40 33 
23 17 13 13 
12 2 1 50 

1 10 50 26 
20 8 23 6 


8 2 24 45 
8 20 46 57 
8 10 2 49 

7 29 18 41 

8 17 40 53 


1 1 6 25 
6 43 31 
10 16 31 36 
8 26 19 41 
8 1 56 47 


5 27 8 52 
7 5 52 53 
7 13 55 40 

7 21 58 27 
9 42 28 


1891 
1892 
1893 
1894 
1895 

1896 
1897 
1898 
1899 
1900 


9 17 11 43 
27 14 44 22 
16 23 32 59 

6 8 21 35 
25 5 54 15 


8 6 56 45 
8 25 18 57 
8 14 34 49 
8 3 50 41 
8 22 12 53 


6 11 44 52 
5 17 21 58 
3 27 10 3 
2 6 58 8 
1 12 34 14 


9 8 45 15 
10 17 2 24 

10 25 32 3 

11 3 34 50 
11 18 51 


13 14 42 52 
2 23 31 28 
21 21 4 9 
11 5 52 44 
30 3 25 24 


8 11 28 45 
8 44 37 
8 19 6 49 
8 8 22 41 
8 26 44 53 


11 22 22 19 

10 2 10 24 

9 7 46 30 

7 17 34 35 

6 23 10 41 


19 21 38 
27 24 25 
2 6 8 26 

2 14 11 13 

3 22 55 14 



The year 1900 will not be leap year : the difference then will be 13 
days between the Old and New Style. 



SEC. XIV.] 



EQ.UATION OF TIME. 

TABLE XVII. 



•159 



Equation of time. 


The Sun faster or slower than the clock. 


Bissextile. 




Jan. 


Feb 


, 


March. 


April. 


May. 


June. 


go 


M. 


s. 


M. 


s. 


M. S. 


M. s. 


M. S. 


M. S. 


1 


4 GO 


8 


I 4 CO 


4 


12 M 27 


3 M 38 

3 g 20 


3 13 


2 m 28 


2 


4 1 


36 


14 p 


11 


12 g 14 


3 g 2 20 


2 g 2 18 


3 


5 r. 


4 


14 i 


17 


12 g 1 


3 I 2 


3 P 27 


2*9 


4 


5 S" 


31 


14 


23 


11 47 


2 44 


3 g» 33 


1 gf 59 


5 


5 3 


58 


14 * 


28 


11 * 32 


2 3 26 


3 3 38 
3^~43 


1 *. 49 

r~7~38~ 


6 


6 n 


25 


14 n 


32 


11 18 


2 9 


7 


6 §> 


50 


14 ^ 


35 


11 ~> 3 


1 "> 51 


3 ^ 47 


1 ~> 27 


8 


7 Mr 


15 


14 ¥? 


37 


10 ~ 47 


1 1-34 


3 §- 51 


1 1- 15 


9 


7 S- 


40 


14 g- 


38 


10 £- 31 


1 & 17 


3 £53 


12-3 


10 


8 ' 


5 


14 * 


39 


10 * 15 


1 ' 1 
45 


3 * 55 


' 51 


11 


8 


29 


14 


39 


9 58 


3 57 


39 


12 


8 


52 


14 


38 


9 41 


29 


3 59 


27 


13 


9 


15 


14 


37 


9 24 


14 


4 


15 


14 


9 


36 


14 


34 


9 7 


2 


4 


2 


15 


9 


57 


14 


31 


8 50 


* 16 


3 59 


* 11 


16 


10 


18 


14 


27 


8 32 


m 31 
e 45 


3 58 


» 24 


17 


10 


38 


14 


23 


8 14 


3 56 


g 36 
S 49 


18 


10 


57 


14 


18 


7 56 


^ 59 


3 53 


19 


11 


16 


14 


12 


7 38 


1 r 12 


3 50 


1 2 


20 
21 


11 


34 


14 


6 


7 20 


1 ~ 25 


3 47 


1 * 16 


11 


50 


13 


58 


7 2 


1 37 


3 43 


1 29 


22 


12 


6 


13 


50 


6 43 


1 -> 48 


3 39 


1 -» 42 


23 


12 


22 


13 


42 


6 25 


1 £59 


3 34 


1 g-55 


24 


12 


37 


13 


33 


6 6 


2 g- 10 


3 29 


2 £■ 8 


25 
26 


12 


51 


13 


24 


5 48 


2 * 21 


3 23 
3 16 


2 ' 20 


13 


4 


13 






2 31 


2 33 


14 


5 29 


27 


13 


16 


13 


3 


r 5 11 


2 41 


3 9 


2 45 


28 


13 


27 


12 


51 


4 52 


2 50 


3 1 


2 57 


29 


13 


38 


12 


39 


4 33 


2 58 


2 53 


3 9 


i 30 


13 


48 






4 15 


3 6 


2 45 


3 20 


Ea 


I 13 


57 




3 56 


2 38 





160 



EQUATION OF TIME. [SEC. XIV. 

TABLE XVII.— Continued. 



Equation of 


time. The Sun faster or slower than the clock. 


Bissextile. 


03 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


M. 


s. 


Iff. s. 


M. s. 


M. S. 


M. 


s. 


M. S. 


1 


1 m 


31 


5 M 50 


V*» ? 4 


10 M 42 


If r* 


15 


10 M 12 


H 


* ft 


42 


5 c 46 


g 53 


11 g 1 


16 g 


lb 


9 c 48 


3 


3 ** 


53 


5'5 41 


1 Z> 12 


11 5, 19 


16 * 


15 


9 S,24 


4 


4 3 


3 


5 S - 35 


1 8 32 


11 if 37 


16 g 


13 


8 g 59 


5 
6 


4 * 


14 

24 


5 * 29 

5 22 


1 ~ 52 


11 ~ 55 


16 t - t 


10 


8 ~ 33 


2 12 


12 12 


16 n 


6 


8 o 7 


7 


4 r» 


33 


5 ^ 14 


2 ^ 32 


12 ~» 28 


16 ~» 


2 


7 ^ 41 


8 


4 £ 


42 


5 1- 6 


2 §- 52 


12 g- 44 


15 ~ 


57 


7 g- 14 


9 


4 g- 


51 


1 g- 58 


3 £ 13 


12 £ 59 


15 & 


51 


6 £-46 


LO 
11 


4 * 


58 


4 * 49 
4 39 


3 ' 34 


13 * 15 


15 * 


44 


6 ' 19 
5 51 


5 


7 


3 54 


13 • 30 


15 


36 


!l.3 


5 


14 


4 29 


4 15 


13 44 


15 


27 


5 22 


113 


5 


21 


4 19 


4 36 


13 58 


15 


18 


4 53 


14 





27 


4 7 


4 57 


14 11 


15 


8 


4 24 


15 

16 


5 


33 


3 55 


5 18 


14 23 


14 


56 


3 54 


5 


39 


3 43 


5 39 


14 35 


14 


44 


3 25 


17 


5 


44 


3 30 


6 


14 47 


14 


31 


2 55 


IS 


5 


49 


3 17 


6 20 


14 56 


14 


18 


2 25 


19 


5 


53 


3 3 


6 41 


15 7 


14 


4 


1 55 


20 





57 


2 49 
2 35 


7 2 


15 16 


13 


49 


1 25 


21 


5 


59 


7 23 


15 25 


13 


32 


55 


22 


6 


1 


2 20 


7 44 


15 33 


13 


15 


25 


23 


6 


3 


2 4 


8 4 


15 40 


12 


58 


5 


24 


6 


4 


1 48 


8 24 


15 47 


12 


40 


* 35 


25 

26 


6 


4 


1 31 


8 45 


15 53 


12 


21 


1 5 


6 


4 


1 14 


9 5 


15 59 


12 


1 


1 g> 35 


27 


6 


4 


57 


9 25 


16 4 


11 


41 


2 § 4 


2S 


6 


2 


40 


9 45 16 8 


11 


19 


2 |- 33 


29 


6 





22 


10 416 10 


L0 


57 


3 3 2 


30 

31 


5 


57 


4 


10 23116 12 


10 


35 


3 a 31 


5 


54 


* 15 




16 14 



SEC. XIV.J EQUATION OF TIME. 

TABLE XVII. -Continued. 



161 



1 "I 


liquation of time. The Sun faster or slower than the clock. 


First after Bissextile. 


w 

1 


Jan. 


Feb. 


March. 

M. S. 


April. 

M. s. 


May. 

M. S. 


June. 


M. S. 


M. S. 


M. S. 


4 M 29 


14 w 10 
14 g 17 


12 


co 30 


3 M 43 
3 W 25 


3 12 


2 no 31 


! 2 


4 g 57 


12 


| 17 


3 g? 19 


2 g 3 22 


3 


5 2 25 


14 £ 22 


12 


5 4 


3 E 6 


3 » 26 


2 » 12 


1 4 


5 o - 52 


14 o 27 


11 


o - 51 


2 0^48 


3 ^32 


2 gf 2 


5 


6 * 19 


14 t* 32 


11 


* 37 


2 3 31 


3 » 37 

3~~42 


1 ~ 52 
l~o~4l" ' 


6 


6 a 44 


14 35 


11 


n 22 


2 13 


/ 


7^9 


14 *? 37 


11 


£ 7 


1 ^ 56 


3 ~» 46 


i r 3i 


8 


7 | 35 


14 i 39 


10 


£52 


1 & 39 


3 ~ 50 


1 §- 19 


9 


7 £59 


14 £ 40 


10 


ft 36 


1 £22 


3 £ 53 


1 £ 7 


10 


8 * 23 


14 * 40 


10 


' 20 


1 • 5 
~49 


3 ' 55 


' 55 


11 


8 46 


14 39 


10 


3 


3 57 


43 


12 


9 9 


14 38 


9 


46 


33 


3 59 


31 


13 


9 31 


14 35 


9 


29 


17 


4 


19 


14 


9 52 


14 32 


9 


12 


2 


4 


7 


15 


10 13 


14 28 


8 


54 


* 13 


4 


0*6 


.16 


10 33 


14 24 


8 


37 


^28 
c 42 


3 59 


m 19 


17 


10 52 


14 19 


8 


19 


3 58 


£ 32 
„, 45 


18 


11 11 


14 14 


8 


1 


^ 56 


3 55 


19 


11 29 


14 8 


7 


43 


1 g 9 


3 52 


o 58 


20 


11 46 
1 


14 


7 


25 


1 " 22 


3 49 


1 * 11 


21 


12 2 


13 52 


7 


6 


1 35 


3 46 


1 o 24 


22 


12 18 


13 44 


6 


48 


1 ~> 47 


3 41 


1 "> 37 


23 


12 33 


13 35 


6 


29 


1 £-58 


3 37 


1 g-50 


24 


12 47 


13 26 


6 


11 


2 £ 9 


3 31 


2 £ 3 


25 

26 


13 1 


13 16 


5 


52 


2 * 19 


3 25 
3 19 


2 * 16 


13 13 


13 6 


5 


34 


2 30 


2 28 


27 


13 24 


12 54 


5 


15 


2 39 


3 12 


2 41 


28 


13 35 


12 42 


4 


56 


2 48 


3 4 


2 53 


29 


13 45 




4 


38 


2 57 


2 56 


3 5 


30 


13 54 


. 4 


19 


3 5 


2 48 


3 17 


31 


1 14 3 


1 4 


1 


2 40 


J 



14* 



162 



EQUATION OP TIME. [SEC. XIV, 

TABLE XVII.— Continued. 



Equation of time 


. The Sun faster or 


slower than the clock. 


First after Bissextile. 


«-< 

CO 

1 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


M. 


s. 


M. 


s. 


M. S. 


M. 


s. 


M. S. 


M. S. 


3 m 


28 


5 m 


51 


M 28 


10 m 


36 


1° ui 1' 


2 


3 R 


39 


5 e 


47 


g 47 


10 s 


55 


16 g 14 


9 = 54 


3 


3 P 


50 


5 S 


42 


1 * 6 


ii i 


13 


16 Z, 13 


9 5, 30 


4 


4 !f 





5 o 


37 


1 1 26 


ii I 


31 


16 g 12 


9 c« 5 


5 
6 


4 * 


11 


5 * 


31 


1 ~ 46 


ii " 


49 


16 w 9 


8 " 39 


1* 


21 


5 n 


24 


2 6 


12 n 


6 


16 o 6 


8 13 


7 


30 


5 r» 


16 


2 ^ 26 


12 "> 


22 


16 ^ 2 


7 ^ 47 


8 


4 & 


39 


5 ~ 


8 


2 ~ 46 


12 £ 


38 


15 £ 58 


7 ~ 20 


9 


4 g- 


48 


5 & 





3 g. 7 


12 & 


54 


15 & 52 


6 & 53 


10 
11 


4 ' 


57 


4 ' 


51 


3 * 27 


13 * 


10 


15 * 45 


6 ' 25 


5 


4 


4 


42 


3 48 


13 


25 


15 37 


5 57 


12 


5 


11 


4 


32 


4 9 


13 


39 


15 29 


5 29 


13 


5 


18 


4 


22 


4 30 


13 


53 


15 20 


5 


14 


5 


25 


4 


10 


4 51 


14 


6 


15 10 


4 3] 


15 

16 


5 


31 


3 


58 


5 12 


14 
14 


19 
31 


14 59 


4 2 


5 


37 


3 


46 


5 33 


14 47 


3 32 


17 


5 


42 


3 


34 


5 54 


14 


42 


14 34 


3 3 


18 


5 


47 


3 


21 


6 14 


14 


53 


14 21 


2 33 


19 


5 


51 


3 


7 


6 35 


15 


3 


14 7 


2 3 


20 


5 


55 


2 
2 


53 

39 


6 56 


15 


13 


13 52 


1 33 


21 


5 


58 


7 17 


15 


22 


13 36 


1 3 


22 


6 





2 


24 


7 37 


15 


30 


13 19 


32 


23 


6 


2 


2 


9 


7 58 


15 


37 


13 2 


2 


24 


6 


3 


1 


53 


8 18 


15 


44 


12 44 


* 28 


25 

26 


6 


4 


1 


36 


8 39 


15 


51 


12 25 


58 


6 


4 


1 


19 


8 59 


15 


56 


12 5 


1 g> 28 


27 


6 


4 


1 


2 


9 19 


16 


1 


11 45 


1 p 57 


28 


6 


3 





45 


9 38 


16 


5 


11 24 


2 |- 26 


29 


6 


1 





27 


9 58 


16 


9 


tl 2 


2 ^ 56 


30 
31 


5 


58 





9 


10 17 


16 


11 


10 40 


3 SL 25 

o 

3 W 53 


5 


55 


* 


9 




16 


13 





SEC. XIV.] 



EQUATION OF TIME. 



163 



TABLE XVII.— Continued. 



Equation of time. The Sun faster or slower than the clock. 


Second after Bissextile. 


d 


Jan. 


Feb. 


M 

M. 


arch. 


April. 


May. 


June. 


M. S. 


M. S. 


s. 


M. s. 


M. S. 


M._ S. 


4 Ul 21 


14 Ul 7 


12 


Ul 32 


3 Ul 46 


3 11 


2 m 34 


2 


4 g 49 


14 g 14 


12 


c 20 


3 f 28 


3 g? 18 


2 g> 25 


3 


5 I 17 


14 S 20 


12 


i e 


3 £ io 


3 » 25 


2 * 15 


4 


5 o 44 


14 o 25 


11 


© 53 


2 o 52 


3 I s 31 


2 g 5 6 


5 


6 3 11 


14 * 29 


11 


* 39 


2 * 34 


3 " 37 


1 " 55 


6 


6 L 37 


14 33 


11 


n 24 


2 16 


1 o « 


7 


7^2 


14 - 35 


11 


- 10 


1 - 59 


3-46 


1 - 34 


8 


7 g-27 


14 g- 37 


10 


£55 


1 £-42 


3 g< 50 


1 £-22 


9 


7 £ 52 


14 g. 39 


10 


g- 39 


1 £-25 


3 g. 53 


1 *- io 


10 


8 * 16 


14 ' 39 


10 


• 23 


1 ' 9 


3 ' 55 


• 58 


11 


8 40 


14 39 


10 


7 


53 


3 57 


46 


12 


9 3 


14 38 


9 


50 


37 


3 59 


34 


13 


9 26 


14 36 


9 


33 


21 


4 


21 


14 


9 47 


14 33 


9 


16 


6 


4 


9 


15 
16 


10 8 


14 30 


8 


59 


0*9 


4 


0*4 


10 28 


14 26 


8 


41 


^24 

c 38 


3 59 


ui 17 


17 


10 48 


14 21 


8 


23 


3 58 


| 30 
„ 43 


18 


11 7 


14 16 


8 


6 


!1 52 


3 55 


19 


11 26 


L4 10 


7 


48 


1 g 5 


3 52 


o 56 


20 
21 


11 43 


14 3 


7 


30 


1 * 18 


3 49 


1 * 9 


11 59 


13 55 


7 


11 




1 o 22 


1 31 


3 46 


22 


12 15 


13 47 


6 


53 


1 -43 


3 42 


1 - 35 


23 


12 30 


13 38 


6 


34 


1 £55 


3 37 


1 g-47 


24 


12 44 


13 29 


6 


16 


2 «. 6 


3 32 


2^0 


26 
26 


12 58 


13 19 


5 


57 


2 * 17 


3 27 
3 21 


2 * 13 


13 10 


13 8 


5 


38 


2 27 


2 25 


27 


13 21 


12 56 


5 


19 


2 37 


3 14 


2 37 


28 


13 32 


12 44 


5 


1 


2 47 


3 7 


2 49 


29 


13 42 




4 


42 


2 56 


2 59 


3 1 


30 


13 52 


4 


23 


3 4 


2 51 


3 13 


31 


14 


4 


5 


2 43 


1 



164 



EQUATION OF TIME. [SEC. XIV. 

TABLE XVII.— Continued. 





Equation of time. The Sun faster or 


slower than the clock. 


Second after Bissextile. 


b 

on 

1 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


M. 


s. 


M. 

5~™ 


s. 
52 


M. 


s. 


M. 


s. 


M. 

16 


s. 


M. S. 


3 rr> 


24 


m 


24 


10 ro 


32 


m 14 




10 M 23 


2 


3 fi 


35 


5 fi 


48 


u 5 


43 


10 c 


50 


16 


Fi 14 


9 g 59 


3 


3 p 

° .CO 


46 


5 - u 

«-» en 


43 


i i 


2 


11 i 


9 


16 


5, 14 


9 S, 35 


4 


3 r 


57 


5 5- 


38 


1 ^ 

1 CO 


21 


ii B 


27 


16 


£ 12 


9 g ii 


5 


4 *» 


7 


5 * 


32 


1 ~ 


41 


ii ~ 


44 


16 


~ 9 


8 " 45 


6 


4 o 


IS 


5 o 


26 


2 o 





12 n 


1 


16 


o 6 


8 19 


7 


4 ^ 


28 


5 ~> 


19 


2 "» 


20 


12 r? 


17 


16 


r* 2 


7 P> 53 


8 


4 - 


37 


5 £ 


11 


2 £ 


41 


12 ~ 


33 


15 


£58 


7 g-26 


9 


4 ** 


46 


5 ft 


3 


3 ft 


1 


12 ft 


49 


15 


£52 


6 ft 59 


10 
11 


4 ' 


55 


4 * 


54 


3 * 


21 


13 * 


5 


15 


* 46 


6 ' 31 


5 


3 


4 


45 


3 


42 


13 


20 


15 


38 


6 3 


12 


5 


11 


4 


35 


4 


3 


13 


34 


15 


30 


5 35 


13 


5 


18 


4 


25 


4 


23 


13 


48 


15 


21 


5 6 


14 


5 


25 


4 


14 


4 


44 


14 


2 


15 


11 


4 37 


15 


5 


31 


4 


2 


5 


5 


14 


15 


15 


1 


4 8 


16 


5 


37 


3 


50 


5 


26 


14 


27 


14 


49 


3 39 


17 


5 


42 


3 


38 


5 


47 


14 


38 


14 


37 


3 9 


18 


5 


47 


3 


25 


6 


8 


14 


49 


14 


24 


2 40 


19 


5 


51 


3 


12 


6 


29 


15 





14 


10 


2 10 


20 


5 


55 


2 


58 


6 


50 


15 


10 


13 


55 


1 40 


21 


5 


58 


2 


43 


7 


11 


15 


20 


13 


40 


1 10 


22 


6 





2 


28 


7 


32 1 


15 


28 


13 


23 


40 


23 


6 


2 


2 


13 


7 


53 


15 


36 


13 


6 


10 


24 


6 


3 


1 


57 


8 


13 


1.5 


43 


12 


48 


* 20 


25 


6 


4 


1 


40 


8 


34 


15 


50 


12 


30 


50 


26 


6 


4 


1 


23 


8 


54 


15 


55 


12 


11 


1 % 19 


27 


6 


3 


1 


6 


9 


14 


16 





11 


51 


1 o 49 


28 


6 


2 





49 


9 


34 


16 


5 


11 


30 2 B 18 


29 


6 


1 





31 


9 


53 


16 


9 


11 


8 2 3 48 


30 


5 


58 





13 


10 


13 


16 


11 


10 


46 


3 a 17 

o 


5 


55 


* 


5 




16 


12 


1 


3 ft 46 



SEC. XIV.] 



EQUATION OF TIME. 



165 



TABLE XVII.— Continued. 



Equation of time. The Sun faster or slower than the clock. 


Third after Bissextile. 




Jan. 


Feb. 


March. 


April. 


May. 


June. 


CO 


M. S. 


M. S. 


M. S. 


M. S. 


M. 


s. 


M. S. 


4 03 U 


H m 5 
14 g 12 


12 35 


3 w 51 
3 | 32 


3 


9 


2 » 36 


2 


4 c 42 


12 g 22 


3 op 


16 


2 g 2 27 


3 


5 S 10 


14 g 18 


12 5 9 


3 S 14 


3 » 


23 


2 s 17 


4 


5 cT 38 


14 o 24 


11 o 56 


2 o^ 56 


3 i? 


29 


2^7 


5 


6^5 


14 ^ 29 


11 3 42 


2 * 39 


3 !» 


35 


1 ^ 57 
1~7~46~ 


6 


6 31 


14 33 


11 28 


2 21 


3 e 


40 


J 7 


6 ^ 57 


14 £ 35 


11 ~> 14 


2 ~> 4 


3 - 


44 


1 "> 35 


8 


7 g-22 


14 |- 37 


10 |- 59 


1 cT 47 


3 W 


48 


1 £24 


9 


7 £-47 


14 & 39 


10 g- 43 


1 & 30 


3 g- 


52 


1 9e 12 


10 


8 ' 11 


14 ' 40 


10 * 27 


1 * 14 


3 ' 


54 


1 ' 


11 


8 35 


14 40 


10 11 


58 


3 


56 


48 


12 


8 58 


14 39 


9 54 


42 


3 


58 


36 


13 


9 21 


14 37 


9 37 


26 


3 


59 


24 


14 


9 42 


14 34 


9 20 


10 


4 





12 


15 

16 


10 3 


14 30 


9 3 


0*5 


4 



59 


0*1 
0~^T3~ 


10 24 


14 26 


8 45 


^20 
W 35 


3 


17 


10 43 


14 22 


8 28 


3 


58 


g 26 
M 39 


s 18 


11 2 


14 16 


8 10 


* 49 


3 


56 


19 


11 20 


L4 10 


7 52 


1 S 3 


3 


54 


o 52 


20 


11 38 


14 3 


7 33 


1 T 16 


3 


51 


1 * 5 


21 


11 54 


13 55 


7 15 


1 29 


3 


48 


1 o 17 


22 


12 10 


13 47 


6 56 


1 °41 


3 


44 


1 "> 30 


23 


12 25 


13 39 


6 38 


1 £53 


3 


39 


1 1-43 


24 


12 39 


13 30 


6 19 


2 & 14 


3 


34 


1 £56 


25 
26 


12 53 


13 20 


6 


2 ; 5 


3 
3 


29 
~23 


2 ' 9 


13 6 


13 10 


5 42 


2 25 


2 21 


27 


13 17 


12 59 


f 5 23 


2 35 


3 


16 


2 34 


28 


13 28 


12 47 


5 4 


2 45 


3 


9 


2 46 


• 29 


13 39 




4 46 


2 54 


3 


1 


2 59 


| 30 


13 49 




4 27 


3 2 


2 


53 


3 11 


I 31 


13 58 




4 9 


2 


45 





166 



EQUATION OP TIME. [SEC. XIV. 

TABLE XVII.— Continued. 



Equation of time. The Sun faster or 


slower than the clock. 


Third after Bissextile. 


S? 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


en 


M. 


S. 


M. 


s. 


M. S. 


M. 


s. 


M. S. 


M. S. 


1 


3 m 


22 


5 rn 


53 


M 19 


10 rn 


27 


16 m 13 


10 M 28' 


2 


3 g 


33 


5 c 


49 


£ 38 


10 c 


46 


16 e 14 


10 g 5 


3 


3 2 


44 


5 y 


45 


5, 57 


11 % 


4 


16 S» 14 


9 S,41 


4 


3 o 


55 


5 ST 


40 


i r i6 


11 $ 


22 


16 §S 13 


9 g 17 


5 
l 6 


4 * 


6 


5 * 


34 


1 ^ 36 


11 ? 


40 


16 ~ 10 


8 ~ 51 


4 „ 


16 


5 n 


28 


1 o 55 


11 n 


57 


16 7 


8 25 


i 7 


4 ? 


26 


5 ~> 


21 


2 ^ 15 


12 ^ 


14 


16 ^ 4 


7 ^ 59 


S 


4 ~ 


36 


5 W 


13 


2 g- 36 


12 ~ 


30 


16 1- 


7 1- 33 


9 


4 £ 


45 


5 S- 


5 


2 £ 56 


12 g- 


46 


15 £- 54 


7 g- 6 


10 
il 


4 * 


54 


4 * 


56 


3 ' 17 


13 * 


2 


15 * 48 


6 ' 39 


5 


2 


4 


47 


3 38 


13 


17 


15 41 


6 11 


12 


5 


9 


4 


38 


3 5S 


13 


32 


15 33 


5 43 


13 


5 


16 


4 


27 


4 19 


13 


46 


15 25 


5 14 


14 


5 


23 


4 


16 


4 40 


14 





15 15 


4 46 


15 


5 


29 


4 


4 


5 1 


14 


3 


15 5 


41 17 


16 


5 


35 


3 


52 


5 23 


14 


26 


14 53 


3 48 


17 


5 


40 


3 


40 


5 44 


14 


37 


14 41 


3 18 


18 


5 


45 


f 3 


27 


6 5 


14 


48 


14 28 


2 48 


19 


5 


49 


3 


14 


6 26 


14 


59 


14 15 


2 18 


20 
21 


5 


53 


3 





6 47 


15 


9 


14 


1 48 


5 


56 


2 


45 


7 8 


15 


19 


13 45 


1 18 


22 


5 


58 


2 


31 


7 28 


15 


27 


13 28 


48 


23 


6 





2 


16 


7 49 


15 


35 


13 11 


18 


24 


6 


2 


2 





8 9 


15 


42 


12 53 


* 12 


25 


6 


3 


1 


44 


8 29 


15 


49 


12 35 


42 




26 


6 


3 


1 


27 


8 50 


15 


54 


12 16 


i ™ 12: 


27 


6 


3 


1 


10 


9 10 


16 





11 56 


1 § 42 


2S 


6 


2 





53 


9 29 


16 


4 


11 35 


2 §- 12! 


29 


6 


1 


[0 


35 


9 49 


16 


8 


11 13 


2 3 41 


30 


5 


59 





IS 


10 8 


16 


10 


10 51 


3 S- 11 

t-p 40 


e 


56 


* 







16 


12 




6\ <J 





SEC. XV.] 



EXAMPLES. 



167 



SECTION FIFTEENTH. 



EXAMPLES 

EXAMPLE I. 



Required the true time of new Moon in July, 1832, 
and also whether there was an eclipse of the Sun or not. 



Mean new 
Moon in 

March, 1832. 
5 lunations. 



from Table 7. 



time once eq'td. 
Table 9th. 



Table 10th. 



Table 11th. 



Eq'tn. of the 
Sun's centre. 



Mean new- 
Moon. 



D. H. M. S. 


1 9 14 37 
147 15 40 15 


149 54 52 
1 46 9 


26 23 8 43 
2 12 8 


26 20 56 35 

3 38 


26 20 52 57 
49 


26 20 53 46 
6 



26 20 47 46 



feun's mean 
Anomaly. 



s 


D. M. s. 


8 1 15 
4 25 31 37 


25 32 52 
6 13 55 6 


4 11 37 42 

Argument 2d. 



Moon's mean 
Anomaly. 



2 5 30 22 
4 9 5 2 



6 14 35 24 

40 18 



6 13 55 6 

Argument 3d. 



Sun's mean dist. 
from the Node. 



6 28 53 54 
5 3 21 10 



2 15 04 

Argument 4th. 



Equal to the 27th day of July, 8 hours, 47 minutes, 
and 46 seconds in the morning, at Washington ; the 
true time of new Moon. The Sun being then only 2 
degrees and 14 minutes from the Moon's ascending 
node, was consequently eclipsed. 



168 



BXAMPLES. 



EXAMPLE II. 



" ? H* 



[sec. XV. 



Required the true time of new Moon in May, 1836, 
and whether there will be an eclipse of the Sun or not. 





Mean new Moon 
in March, 1836. 


Sun's mean 
Anomaly. 


Moon's mean 
Anomaly. 


Sun's m. dist. 
from the node. 


D. H. M. 


s. 


S. 


D. M. S. 


s. D. M. g. 


S. D. M. S. 


1836= 
Table 3d. 


16 21 57 
59 1 28 


10 
6 


8 
1 


15 17 23 

28 12 39 


9 6 20 44 
1 21 38 1 


10 2 27 30 
2 1 20 28 


Table 7th. 


14 23 25 
2 54 


16 
24 


10 
10 


13 30 2 

29 6 47 


10 27 58 45 
1 8 2 


3 47 58 


Table 9th. 


15 2 19 
5 21 


40 
56 


11 


14 23 15 10 29 6 47 








Table 10th. 


14 20 57 
1 


44 
20 


• :- - 


Table 11th. 


14 20 59 


4 
13 


Table 17th. 


14 20 59 
4 


17 
1 




14 21 3 


18 









Equal to the 15th day of May, 9 hours, 3 minutes, and 
18 seconds ; true time of new Moon at Washington. 
The Sun being then only 3 degrees and 48 minutes from 
the Moon's node, the Sun will consequently be visibly 
eclipsed. 



SEC. XV.] 



EXAMPLES. 
EXAMPLE III. 



169 



Required the true time of new Moon, in December, in 
the year 1850 ; and whether there will be an Eclipse at 
that time or not. 



Mean New Moon 
in March, 1850. 



D. 


H. 


M. 


s. 


12 

265 


16 

18 


57 
36 


57 

27 


3 


11 

1 


34 

58 


24 
3 


3 


9 

9 


36 
11 


21 
13 


3 
3 






25 
2 

28 
1 


8 
54 

2 
27 


3 





26 
9 


35 

42 



36 17 



Sun's mean 
Anomaly. 



10 31 11 
21 56 54 



2 28 5 
24 15 25 



Moon's mean 
Anomaly. 



2 2 38 59 

7 22 21 4 

9 25 3 

44 38 



8 12 409 24 15 25 



Sun's mean 

distance from 

the node. 



s. 


D. M. 


s. 


6 
9 


28 32 
6 2 


38 
6 


4 


4 34 


44 





True time of new Moon in December, 1850, will be 
the 3d day, 36 minutes, 17 seconds, afternoon. The 
Sun will then be more than 56 degrees from the node, 
and consequently there can be no eclipse at that time, 



15 



170 



EXAMPLES. 



[SEC. XV. 



EXAMPLE IV. 



Required the true time of full moon in July 1833, and 
whether or not there will be an eclipse of the Moon at 
that time. 



Mean new 
Moon in March. 



D. H. 


M. 


s. 


20 6 
38 14 
14 18 


47 
12 

22 


17 

9 
2 


1 15 


21 
1 


28 

7 


1 15 

7 


20 

42 


21 

8 


1 7 


38 
3 


13 

38 


1 7 


34 


35 
16 


1 7 


34 
3 


19 

22 



1 7 37 41 



Sun's mean 
anomaly. 



s. 


D. 


M. 


s. 


8 
2 



18 
27 
14 


23 

18 
33 


'27 
58 
10 



10 



11 


15 

28 


35 
34 



1 18 47 1 



Moon's mean 
anomaly. 



s. 


D. M. 


s. 


1 

2 
6 


11 7 
17 27 

12 54 


28 

1 

30 


10 


11 28 


59 
25 



10 11 28 34 



Sun's mean 

distance from 

the node. 



S. D. 


M. 


8. 


8 7 
3 2 
15 


37 



20 


55 
42 

7 


11 24 


58 


44 



in the afternoon, 



Equal to the 1st day of July, 1833, the true time of full 
Moon in the longitude of Washington, at 7 hours, 37 
minutes 'and 41 seconds in the afternoon, the Sun being 
then within five degrees at a mean rate from the Moon's 
node, consequently the Moon was then visibly eclipsed. 



EXAMPLE v. 



Required the true time of full Moon in April, in the 
year 1836 at Rochester ; and also, whether there will be 
an eclipse of the Moon, or not. 

The true time of full Moon, in April, in the year 
1836, will be on the first day, 5 hours 19 minutes and 



SEC. XV.] 



EXAMPLES. 



171 



53 seconds in the afternoon, in the longitude of Roches- 
ter ; the Sun will then be more than 40 degrees from the 
Moon's node, and consequently there will be no eclipse 
on that day. 



EXAMPLE VI. 



Required the true time of mil Moon, in September in 
the year 1848, in the longitude of Utica ; and whether 
there will be an eclipse at that time. 



Mean time of full 
Moon in March. 



D, 



4 

177 

14 



12 



M. 6. 



10 36 41 
4 24 18 

18 22 2 



9 23 1 
3 57 2 



12 


5 

7 


25 

48 


59 
40 


12 


13 


14 
3 


39 
46 


12 


13 


18 


25 
5 


12 


13 


18 
4 


30 

8 

38 


12 


13 


22 
6 


12 


13 


28 


38 



Sun's mean 
Anomaly. 



M. 



8 2 53 7 
5 24 37 56 
14 33 10 



12 4 13 
3 32 30 



10 8 31 43 



Moon's mean Sun's mean dist 
Anomaly. from the node, 



s. 


D- 


M. 


s. 


4 


17 


13 


48 


5 


4 


54 


3 


6 


12 


54 


30 


4 


5 


2 


21 




1 


29 


51 



True time at Lyons. 
True time at Utica. 



s. 


D. 


M. 


s. 


5 


11 


45 


53 


6 


4 


1 


24 





15 


20 


7 





1 


7 


21 



4 3 32 30 



The Moon will be full in the year 1848, on the 13th 
day of September, at 1 o'clock and 28 minutes in the 
morning, in the longitude of Utica ; the Sun, then, will 
be only 1 degree and 7 minutes from the Moon's node ; 
the Moon therefore will be eclipsed at that time. 



172 EXAMPLES. [SEC. XV. 

To calculate the true time of any new, or full Moon, 
and consequently eclipses, in any given year and month, 
between the commencement of the Christian Era, and 
that of the 18th century. 

Find a year of the same number in the 18th century, 
with that of the year in the proposed century, from Table 
1st, and take out the mean time of new Moon in March, 
Old Style, for that year, with the mean anomalies of 
the Sun and Moon, and the Sun's mean distance from 
the node at that time, as before instructed. Take as 
many complete centuries of years from Table 2d as when 
subtracted from the year of the 18th century, the remain- 
der will answer to the given year, with the anomalies, 
and Sun's distance from the node ; subtract these from 
those of the 18th century, and the remainder will be the 
mean time of new Moon in March, with the anomalies } 
&c. for the proposed year ; then proceed, in all respects } 
for the true time of new or full moon, as shown in the 
Precepts, or former Examples. 

If the days annexed to these centuries, exceed the 
number of days from the beginning of March, taken out 
in the 18th century, subtract a lunation, and its anoma- 
lies, &c. from Table 3d, from the time, and anomalies 
of new Moon in March, and then proceed as above stated, 
This circumstance happens in Example 5th. 



SZC. XV.] 



EXAMPLES. 



173 



EXAMPLE VII. 

Required the true time of new Moon in June, in the 
year of Christ, 36, at the city of Jerusalem. 





Mean N. Moon 
in March. 


Sun's mean 
Anomaly. 


Moon's mean 
Anomaly. 


Sun's mean 

dist. from the 

Node. 


By the.Precepts. 


D. 


H. M. S. 


D. D. M. s. 
S. D. M. S. 


S. D. M. s. 


S. D. M. s. 


March, 1736. 
Add one lunation. 



9 


18 54 2 
12 44 3 


S 11 52 22 
29 6 19 


8 20 58 49 
25 49 


5 12 24 27 
1 9 40 14 


Lastn. Moon, March, 1736. 
Subtract 1700 years. 



9 


7 33 5 
19 U 25 


9 10 58 41 

8 19 58 48 


1 16 47 49 
1 22 30 37 


6 13 34 41 
6 14 31 7 


Mean n.Moon,Mar.in36. 
Add three lunations. 


21 

38 


15 26 40 
14 12 9 


20 59 53 
2 27 18 58 


11 24 17 12 
2 17 27 I 


21 29 3 34 
3 2 42 


By Table Fourth= 


110 








June 

First equation 


18 


5 38 49 
3 54 43 


3 18 18 50 
Ar. for 1st eq. 


2 11 44 13 
1 30 55 


3 1 4 16 




18 


1 39 00 
9 24 15 


3 18 18 51 
2 10 13 18 

1 8 5 33 
Arg.3d. eq. 


2 10 13 18 
Arg. 2d eqt. 


3 1 4 16 

Arg. 4th eqt. 


Third equation. . . . 


18 
18 


11 3 21 
2 54 






11 28 
3 




True time of new Moon at 
London. 


18 


11 24 
2 20 


True time at Jerusalem. 


13 


13 20 24 









The true time of new Moon in June, in the year of 
our Lord 36, on the 19th day, at 1 hour, 20 minutes 
and 24 seconds, in the morning ; the mean distance of 
the Sun being 3 signs, 1 degree, 4 minutes, and 16 sec- 
onds from the Moon's ascending node, consequently 
there was no eclipse at that time. 

To calculate the true time of new, or full Moon, and 
also to know whether there will be an eclipse at the 
time, in anv given year and month before the Christian 
Era. 

15* 



174 PRECEPTS AND EXAMPLES. [SEC. XV. 

Find a year in the 18th century from Table 1st, which 
being added to the given number of years before Christ 
diminished by one, shall make a number of complete 
centuries. Find this number of centuries in Table 2d, 
and subtract the time, anomalies and distances from the 
node belonging to it, from those of the mean new Moon 
in March, the above found year, in the 18th century, and 
the remainder will denote the time, anomalies, &c. of 
the mean new Moon in March, the given year, before 
Christ ; then for the true time thereof in any month of 
that year, proceed as before directed. 



SEC. XV. 



PRECEPTS AND EXAMPLES. 



175 



EXAMPLE VIII. 

Required the true time of new Moon in May, Old Style, the year 
before Christ, 585, at Alexandria, in Egypt. The year 584, added to 
1716, make 2300, or 23 centuries. 



March, 1716. 

March, 2000. 
Do. 300. 



Subtract 1 lunation. 



New Moon, 2300, 



Which subtract from 1716 
March, B. C. 585. 
Add 3 lunations. 

New Moon March, 585. 
First equation. 



Second equation. 



Third equation. 



Fourth equation. 



Clock slower. 



Time at London. 
Difference of longitude. 



May. 



Mean n.Moon Sun's mean 
in March. Anomaly. 



11 17 33 29 



27 18 9 19 
13 32 37 



40 18 52 56 
29 12 44 3 



11 5 58 53 



11 34 36 
18 14 12 9 



28 1 46 45 
1 37 



28 1 45 



28 1 45 08 
2 15 1 



S. D. M. S. 



8 22 50 39 



8 50 
10 3 



18 53 
29 6 19 



11 19 46 41 



9 3 03 58 
2 27 18 58 



Moon's mean 
Anomaly 



4 4 14 2 



15 42 

1 16 6 



2 1 48 
25 49 



1 5 59 



2 28 15 2 
2 17 27 1 



22 56 

Arg. 1st eq. 



2S 


4 00 9 
1 9 


23 


4 01 18 
12 


28 


4 1 30 
3 


23 


4 4 30 
2 2 



28 6 6 30 



22 56 
5 15 41 17 


6 14 41 39 
Arg. 3d eqn. 


6 14 41 39 



5 15 42 3 
46 



5 15 41 17 
Arg. 2d eq, 



5 15 41 17 



5 15 41 17 



sun's mean 
dist. from 
the Node. 



4 27 r 



6 27 45 

1 28 22 



3 26 07 
1 40 14 



7 25 26 46 



9 1 50 19 
3 2 42 



3 51 01 



3 51 01 



3 51 1 



3 51 1 
Arg.4theq. 



The true time of new Moon at Alexandria in May, 585 years 
before Christ, was on the 28th day, 6 hours, 6 minutes and 30 seconds, 
afternoon. The Sun being then only 3 degrees and 51 minutes from 
the Moon's ascending node, was consequently eclipsed. 

The above eclipse was central, and total in North America at 
eleven o'clock in the morning; it also passed centrally over the 
south parts of France and Italy. The duration of total darkness 
being about 3 minutes: 



176 



PRECEPTS AND EXAMPLES. [SEC XV. 



EXAMPLE IX. 



Required the true time of full Moon at Alexandria 
in Egypt, in September, Old Style, in the year 201 
before the Christian Era. 200 years added to 1800, 
make 2000, or 20 centuries. 



By the Precepts. 


Mean new 
Moon in 
"March. 


Sun's mean 
anomaly. 


Moon's 

mean 

aliomaly. 


Sun's dist. 
fr. Moon's 
ascen.node 




D. H. M. s. 


S. D. M. s. 


S. D. M. s. 


S. D. M. s. 


March, 1800. 
Add one lunation. 


13 22 17 
29 12 44 3 


8 23 19 55 
29 6 19 


10 7 52 36 
25 49 


11 8 58 24 
1 40 14 


From the same. 
Subtract 2000 years. 


42 13 6 20 
27 18 9 19 


9 22 26 14 
8 50 


11 3 41 36 
15 42 


4 38 38 
6 27 45 


Mean n. Moon, B. C. 201 
Add six lunations. 


14 19 57 01 
177 4 24 18 


9 13 36 14 
5 24 37 56 


10 17 59 36 
5 4 54 3 


5 6 53 38 

6 4 1 24 




191= 








N. Moon, Sep. B. C. 201 
Add one half lunation. 


7 23 21 19 
14 18 22 2 


3 8 14 10 
14 33 10 


3 22 53 39 
6 12 54 30 


11 10 55 2 
15 20 7 


Full Moon Sept. 201. 
First equation. 


22 17 43 21 
3 52 6 


3 22 47 20 
Arg. 1st eqt. 


10 5 48 9 
1 28 14 

10 4 19 55 
Arg. 2d eqt. 


11 26 15 9 


Time once equated. 
Second equation. 


22 13 52 15 

8 25 4 


3 22 47 20 
10 4 19 55 


11 26 15 9 


Time twice equated. 
Third equation. 


22 5 26 11 
58 

22 5 25 13 
12 


5 18 27 25 
Arg. 3d eqt. 


10 4 19 55|ll 26 15 9 
Arg. 4th eqt. 






True time at London. 


22 5 25 1 


Add for diff of long. 


2 2 


Add a Clock. 


22 7 27 1 
7 33 


Tree time at Alexan- 
dria. 


22 7 34 34 









The true time of full Moon, at Alexandria in Egypt 
in the year before Christ 201, in September, was on the 



SEC. XV.] PRECEPTS AND EXAMPLES. 177 

22d day, 7 hours, 34 minutes, and 34 seconds, the actual 
time of opposition. The Sun being within three degrees 
and 45 minutes of the Moon's ascending node, conse- 
quently the Moon was visibly eclipsed at that time at 
Alexandria. 

To calculate the true time of new or full Moon, and 
Eclipses in any given year ; and month after the 18th 
century. 

Find a year of the same number in the 18th century 
with that of the year proposed, and take out the mean 
time, and anomalies &c. of new Moon in March, Old 
Style, from Table 1st for that year. 

Take so many years from Table 2d, as when added 
to the above mentioned year in the 18th century will 
answer to the given year in which the new or full Moon 
is required ; and take out the first new Moon with its 
anomalies (fee. for these complete centuries. Add all 
these together, and then proceed as before directed, to 
reduce the mean to the true syzygy. It is however 
necessary to remember to subtract a lunation with its 
anomalies, when the above said addition carries the new 
Moon beyond the 31st day of March, as in the following 
example. 






178 



PRECEPTS AND EXAMPLES. [SEC. XV. 



EXAMPLE X. 



Required the true time of new Moon in July, Old 
Style, 2180 at Washington. 



FOUR CENTURIES ADDED TO 1780 


MAKE 2180. 






Mean new- 
Moon in 
March. 


Sun's mean 
anomaly. 


Moon's 

mean 

anomaly. 


Sun's dist. 
fr. Moon's 
ascen.node 


By the Precepts. 


D. H. M. s. 


S. D. M. s. 


S. D. M. s. 


S. D. M. s. 


March, 1780. 
Add 4oo years. 


23 23 1 44 
17 8 43 29 


9 4 18 13 
13 24 


1 21 7 47 
10 1 28 


10 18 21 1 
6 17 49 


Subtract 1 lunation. 


41 7 45 13 
29 12 44 3 


9 17 42 13 
29 6 19 


11 22 35 47 
25 49 


5 6 10 1 
1 49 14 


Mean time new Moon, 
March, 2180 add four 
lunations. 


11 19 1 10 
118 2 56 12 

129 


8 18 35 54 
3 26 25 17 


10 26 46 47 
3 13 16 2 


4 5 29 47 
4 2 40 «56 


New Moon, July, 2180 
First equation. 


7 21 57 22 
1 3 39 


15 1 11 
Arg. 1st eqt. 


2 10 2 49 1 , 8 8 10 43 
24 12 J 


Time once equated. 
Second equation. 


7 20 53 43 
9 24 8 


15 1 11 
2 9 38 37 


2 9 38 37 
Arg. 2d eqt. 


8 8 10 43 
Arg. 4th eqt. 


Third equation. 


8 6 17 51 
3 56 


10 5 22 34 
Arg. 3d eqt. 








Fourth equation. 


8 6 21 47 
1 8 








F. Clock. 


8 6 22 55 
4 30 


True time at London. 
Difference of long. 


8 6 18 25 
5 8 


True time atWashing'n 


8 1 10 25 









The true time of new Moon, Old Style, will then be 
on the 8th day of July, 1 hour 10 minutes and 25 sec- 
onds afternoon ; or the 22d day, at the same hour, min- 
ute and second, New Style. 



SEC. XV.] PRECEPTS AND EXAMPLES. 



179 



EXAMPLE I. 



For finding the Sun's true place. 

Required the Sun's true place, March 20th, 1764, Old 
Style, at 22 hours, 30 minutes, 25 seconds past noon, 
In common reckoning, March 21st, at 10 hours 30 min- 
utes and 25 seconds in the morning, 



BY THE PRECEPTS. 


Sun's long. 

S. D. M. S. 


Sun's anom. 

S. D. M. s. 


To the radical year after Christ, . . . 1701 
Add complete years, 60 

o 


9 20 43 50 

27 12 
11 29 17 

1 28 9 11 
20 41 55 

54 13 

1 14 

1 


6 13 1 

11 29 26 

11 29 14 

1 28 9 

20 41 55 

54 13 

1 14 


Bissextile days, 20 

Hours, . 22 

Minutes, 30 


Seconds, 25 


Sun's mean place at the given time, 

Add eq'tn of the Sun's centre, from table 6, 


10 14 36 
1 55 36 

12 10 12 


9 1 27 23 


The argm't. 
with which 
enter table 6. 


Sun's true place, .....' 


That is Aries, 12 deg., 10 minutes, 12 sec. 



180 



ECLIPSES OP THE SUN AND MOON. [SEC. XV. 



EXAMPLE II. 

Required the Sun's true place, October 23, Old Style, 
at 16 hours, 57 minutes past noon in the 4008th year 
before Christ ; which was the 4007th year before the 
year of his birth, and the year of the Julian period, 706. 
This is supposed by some to be the very instant of the 
creation. 



BY THE PRECEPTS. 


Sun's long. 


Sun's anom. 

S. D. M. B. 


S. D. M. s. 


From the radical number after Christ, . . 
Subtract for 5000 complete years, .... 


9 7 53 10 
1 7 46 40 


6 28 48 
10 13 25 




8 6 30 
6 48 
36 16 
5 26 
8 29 4 54 
22 40 12 
39 26 
2 20 


8 15 23 
11 21 37 
11 29 15 
11 29 15 

8 29 4 

22 40 12 

39 26 

2 20 


To which add ( 900 


Complete years, \ 80 

October, 

Days, • . 

Hours, 


Minutes, 


Sun's mean place at the given time, . . . 
Subtract equation of the Sun's centre, . . 


6 3 4 
3 4 


5 28 33 58 

Arg't eq'tn. 
Sun's centre 


Sun's true place at that time, . 

Which was just entering the sign Libra. . 


6 



CONCERNING ECLIPSES OF THE SUN AND MOON. 

To find the Sun's true distance from the Moon's 
ascending node, at the time of any given new or full 
Moon, and consequently to know whether there be an 
eclipse at that time or not. 

The Sun's mean distance from the Moon's ascending 
node, is the argument for finding the Moon's fourth equa- 
tion in the syzygies, and therefore it is taken in all the 
foregoing Examples, in finding the true times thereof. 
Thus at the time of mean new Moon in March, 1764, 
Old Style, or April in the new, the Sun's mean distance 
from the ascending node is signs, 35 minutes, 2 sec- 
onds. [See Example 11th.] The descending node is 
opposite to the ascending one, and consequently they are 



SEC. XV.] ECLIPSES OP THE StiN AND MOON. 181 

exactly 6 signs distant from each other. When the Sun 
is within 17 degrees of either of the nodes at the time 
of new Moon, he will be eclipsed at that time, as before 
stated ; and at the time of full Moon, if the Sun be within 
12 degrees of either node, she will be eclipsed. Thus 
we find from Table 1st, that there was an eclipse of the 
Sun. at the time of new Moon, April 1st, at 30 minutes, 
25 seconds after 10 in the morning, at London, New 
Style, when the old is reduced to the new, and the mean 
time reduced to the true. 

It will be found by the Precepts, that the true time of 
that new Moon is 50 minutes, 46 seconds later, than the 
mean time, and therefore we must add the Sun's motion 
from the node during that interval to the above mean 
distance signs, 6 degrees, 35 minutes, 2 seconds, which 
motion is found, in liable 12th, for 50 minutes and 46 
seconds, to be 2 minutes, 12 seconds, and to this apply 
the equation of the Sun's mean distance from the node in 
Table 13th, which at the mean time of new Moon, April 
1st, 1764, is 9 signs, 1 degree, 26 minutes, and 20 sec- 
onds, and we shall have the Sun's true distance from the 
node at the true time of new Moon, as follows : 

Sun from node, 

s. d. m. s. 

At the mean time of new Moon in April , 1 764, 5 352 

Sun's motion from node for 50 minutes, 2 10 

For 46 seconds, 2 



Suns mean dist.from node at true new Moon, 5 37 14 
Equation from mean dist. from node, add, 2 5 



Sun's true dist from the ascending node, 7 42 14 
which being far within the above named limits of 17 
degrees, the Sun was at that time eclipsed. The man- 
ner of projecting this or any other eclipse, either of the 
Sun or Moon, will now be shown. 



16 



182 ELEMENTS FOR SOLAR ECLIPSES. [SEC. XVI. 



SECTION SIXTEENTH. 



TO PROJECT AN ECLIPSE OF THE SUN. 

To project an Eclipse of the Sun, we must from the 
Tables find the ten following Elements : 

1st. The true time of conjunction of the Sun and 
Moon, and 

2d. The semidiameter of the earth's disk, as seen 
from the Moon, at the true time of conjunction, which is 
equal to the Moon's horizontal parallax. 

3d. The Sun's distance from the solstitial colure, to 
which he is then nearest. 

4th. The Sun's declination. 

5th. The angle of the Moon's visible path with the 
ecliptic. 

6th. The Moon's latitude. 

7th. The Moon's true horary motion from the Sun. 

8th. The Sun's semidiameter. 

9th. The Moon's semidiameter. 

10th. The semidiameter of the penumbra. 



SEC. XVI.] DELINEATION OF SOLAR ECLIPSES. 



183 



EXAMPLE XI. 

Required the true time of new Moon at London, kx 
April, 1764, New Style, and also whether there were an 
eclipse of the Sun or not at that time ; and likewise the 
elements necessary for its protraction, if there were at 
that time an eclipse. 



By the Precepts. 


Mean time of 

new Moon in 

March. 


Sun's mean 
anomaly. 


•.«■ , 1 Sun's mean 

a °nnS a w ean dislancefrom 
anomaly. | the node . i 




D. H. M. S. 


S. D. M. S.| S. D. M. S. | S. D. M. S. 


March, 1764 
Add 1 lunation 


2 8 55 36 
29 12 44 3 


8 2 20 
29 6 19 


10 13 35 21 
25 49 


11 4 54 48 
1 40 14 


Mean n. Moon 
First equation 


31 21 39 39 
4 10 40 


9 1 26 19 

Arg. lsteqa. 


11 9 24 21 
1 34 57 


5 35 2 

i 


Second equa. 


32 1 50 19 
3 24 49 


9 1 26 19 
11 10 59 18 


11 10 59 18 

Arg. 2d eqa. 




Third equation 


31 22 25 30 
4 37 


9 20 27 1 
Arg. 3d eq. 


11 10 59 18 


5 35 2 

Arg.4theq. 


True n. Moon 
Equa. of days 


31 22 30 7 

18 

31 22 30 25 

3 48 






Sun from 

node. 
5 35 2 




31 22 26 35 


] 





The true time is April 1st, 10 hours, 26 minutes, 35 
seconds in the morning, tabular time. The mean dis- 
tance of the Sun at that time being only 5 degrees, 35 
minutes and 2 seconds past the ascending node, the Sun 
was at that time eclipsed. Now proceed to find the ele- 
ments necessary for its protraction. The true time being 
found as above. 

To find the Moon's horizontal parallax, or semidiam- 
eter of the Earth's disk as seen from the Moon. Enter 
Table 15th with the signs and degrees of the Moon's 
anomaly, (making proportion because the anomaly is in 
the table calculated only to every 6th degree,) and from 
it take out the Moon's horizontal parallax, which for the 
above time is 54 minutes and 53 seconds, answering to 
the anomaly of lis. 9d. 24m. 21 seconds. 

To find the Sun's distance from the nearest solstice, 



184 DELINEATION OF SOLAR ECLIPSES. [SEC. XVI. 

namely, the beginning of Cancer, which is 3 signs, or 90 
degrees from the beginning of Aries. It appears from 
Example 1st for calculating the Sun's true place, the cal- 
culation being made for the same time, that the Sun's lon- 
gitude from the beginning of Aries, was then 0s. 12d. 
10m. 12 seconds ; that is, the Sun's place was then in 
Aries," 12 degrees, 10m. 12 seconds, therefore from 

s. d. m. s. 

3 
Subtract the Sun's longitude or place, 12 10 12 



Remains Sun's distance from the solstice, 2 17 49 48 



which is equal to 77 degrees, 49 minutes, 48 seconds, 
each sign containing 30 degrees. 

To find the Sun's declination, enter Table 5th, with 
the signs and degrees of the Sun's true place, namely, 12 
degrees, 10m. 12s. and making proportions for the 10m. 
12 seconds, takeout the Sun's declination, answering to 
his true place, and it will be found to be 4 degrees 49 
minutes north. 

To find the Moon's latitude. This 'depends on her true 
distance from her ascending node, which is the same as 
the Sun's true distance from it at the time of new Moon, 
and is thereby found in Table 14th. But we have already 
found, [see Example,] by calculating the Sun's true 
place, that, at the true time of new Moon in April, 1764, 
the Sun's equated distance from the node was 0s. 7d. 
42m. 14s. ; therefore enter Table 14th with the above 
equated distance, (making proportions for the minutes and 
seconds,) her true latitude will be found to be 40 minutes 
and 18 seconds north ascending. 

To find the Moon's horary motion from the Sun. With 
the Moon's anomaly, namely, lis. 9d. 24m. 21s., enter 
Table 15th, and take out the Moon's horary motion, 
which, by making proportions in that table, will be found 
30 minutes 22 seconds. Then with the Sun's anomaly, 
namely, 9s. Id. 26m. 19s. (in the present case,) take out 
this horary motion, 2 minutes and 28 seconds from the 



SEC. XVI.] DELINEATION OF SOLAR ECLIPSES. 185 

same table ; subtract the latter from the former, and the 
remainder will be the Moon's horary motion from the 
Sun ; namely, 27 minutes and 54 seconds. 

To find the angle of the Moon's visible path with the 
ecliptic. This, in the projection of eclipses, may be al- 
ways rated at 5 degrees and 35 minutes, without any 
sensible error. 

To find the semidiameters of the Sun and Moon. — 
These are found in the same table, (15,) and by the same 
Argument, as their horary motions. In the present case, 
the Sun's anomaly gives his semidiameter 16 minutes 
and 6 seconds, and the Moon's anomaly gives her diam- 
eter 14 minutes and 27 seconds. 

To find the semidiameter of the penumbra. Add the 
Sun's semidiameter to the Moon's, and their sum will be 
the semidiameter of the penumbra ; equal to 31 minutes 
and 3 seconds. 

Collect these elements together, that they may the 
more readily be found when they are wanted in the con- 
struction of this eclipse. Thus : — 

D. H. M. S. 

1st. The true time of new Moon in April, 1 10 30 25 



2d. Semidiameter of the earth's disk, 

3d. Sun's distance from the nearest solstice, 

4th. Sun's declination north, 

5th. Moon's latitude, north, ascending, 

6th. Moon's horary motion from the Sun, 

7th. Angle of Moon's visible path with ecliptic, 

8th. Sun's semidiameter, 

9th. Moon's semidiameter, 

10th. Semidiameter of the penumbra, 

To project an eclipse of the Sun geometrically.— 
Make a scale of any convenient length, A. C. and divide 
it into as many equal parts as the earth's semi-disk con- 
tains minutes of a degree ; which at the time of the 
eclipse in April, 1764, was 54 minutes and 53 seconds ; 
then with the whole length of the scale as a radius, de- 

16* 



D. M. 


s. 


54 


53 


$. 77 49 48 


4 49 





40 


18 


27 


54 


)tic, 5 35 





16 


6 


14 


57 


31 


3 



186 DELINEATION OP SOLAR ECLIPSES. [SEC. XVI. 

scribe the semicircle A M B upon the centre C, which 
semicircle will represent the northern half of the earth's 
enlightened disk, as seen from the Sun. 

Upon the centre C, raise the straight line C H perpen- 
dicular to the diameter A C B, then will A C B be a 
part of the ecliptic, and C H its axis. 

Being provided with a good sector, open it to the ra- 
dius C A in the line of chords, and taking from thence the 
chord of 23 degrees and 28 minutes in your compasses, 
set it off both ways from H to g, and g to A, in the peri- 
phery of the semi-disk, and draw the straight line g V A, 
in which the north pole of the disk will be always found. 

When the Sun is in Aries, Taurus, Gemini, Cancer, 
Leo and Yirgo, the north pole of the earth is enlightened 
by the Sun ; but while the Sun is in the other six signs, 
the south pole is enlightened. 

When the Sun is in Capricorn, Aquarius, Pisces, Aries, 
Taurus, and Gemini, the northern half of the earth's 
axis, C XII P lies to the right hand of the axis of the 
ecliptic, as seen from the Sun ; and to the left hand, 
whilst the Sun is in the other 6 signs. 

Open* the sector, till the radius [or distance of the two 
90s] of the signs be equal to the length of V h, and take 
the sign of the Sun's distance from the solstice, 77 degrees, 
49 minutes, and 48 seconds, in your compasses from 
the line of sines, and set off that distance from V to P 
in the line of g V h, because the earth's axis lies to the 
right hand of the axis of the ecliptic in this case, [the 
Sun being in Aries,] and draw the straight line C XII 
P for the earth's axis, of which P is the north pole. If 
the earth's axis had lain to the left hand from the axis of 
the ecliptic, the distance V P would have been set off 
from V towards g. 

* To persons acquainted with Trigonometry, the angle contained 
between the earth's axis and that of the ecliptic, maybe found more 
accurately by calculation. 

Rule. — As radius is to the sine of the Sun's distance from the sol- 
stice, so is the tangent of the distance of the poles (23 degrees and 
28 minutes) to the tangent of the angle contained by the axis. Then 
set off the chord of the angle from H to h, and join C H, which 
will cut F G in P, the place of the north pole. 



SEC. XVI.] DELINEATION OF SOLAR ECLIPSES. 187 

To draw the parallel of latitude of any given place, 
as suppose for London in this case, or the path of that 
place on the earth's enlightened disk, as seen from the 
Sun, from sunrise to sunset, take the following method. 

Subtract the latitude of London in this case, 51 de- 
grees and 30 minutes, from 90 degrees, and the remain- 
der, 38 degrees and 30 minutes, will be the co-latitude, 
which take in your compasses from the line of chords, 
making A or C B the radius, and set it from h to the 
place where the earth's axis meets the periphery of the 
disk to VI and VI, and draw the occult or dotted line 
VI K VI, then from the points where this line meets 
the earth's disk, set off the chord of the Sun's de- 
clination, 4 degrees and 49 minutes, to E and F, and 
to E and G, and connect these points by the two occult 
lines, F XII G and E L E. 

Bisect L, K, XII, in K, and through the point K draw 
the black line VI K VI, then making C B the radius 
of a line of sines on the sector, take the co-latitude of Lon- 
don, (38i degrees,) from the sines in your compasses, 
and set it both ways from K to VI and VI. These hours 
will be just in the edge of the disk at the equinoxes, but 
at no other time in the whole year. With the extent K 
VI taken into your compasses, set one foot in K in the 
black line below the occult one as a centre, and with the 
other foot describe the semicircle VI, 7, 8, 9, 10, &c. and 
divide it into 12 equal parts ; then from these points of 
division draw the occult 7, p, 8, 0, 9, ?i, parallel to the 
earth's axis, C, XII, P. 

With the small extent K XII as a radius, describe the 
quadrantal arc XII /, and divide it into six equal parts, 
as XII, a, a, b, be, cd, de, ef, and through the division 
points a, b, c, d, e, draw the occult lines VII, e, V, VIII, 
d, IV, IXC, III, X, 6, II, and XI, a, I, all parallel to VI, K, 
VI, and meeting the former occult lines 7, p, 8, 0, &c. in 
the points VII, VIII, IX, X,XI, V, IV, III, II, and I, which 
points will mark the several situations of London on the 
earth's disk at these hours respectively, as seen from the 
Sun ,and the elliptic curve VI, VII, VIII, &c. being drawn 



188 DELINEATION OF SOLAR ECLIPSES. [SEC. XVI. 

through these points, will represent the parallel of lati- 
tude, or path of London, on the disk as seen from the 
Sun from its rising to its setting. 

If the Sun's declination had been south, the diurnal 
path of London would have been on the upper side of 
the line YI K VI, and would have touched the line D L 
E in L. It is necessary to divide the hourly spaces into 
quarters, and if possible into minutes also. 

Make C B the radius of a line of chords on the sector, 
and taking therefrom the chord of 5 degrees and 35 
minutes, (the angle of the Moon's visible path with the 
ecliptic ;) set it off from H to M on the left hand of C H, 
(the axis of the ecliptic) because the Moon's latitude in 
this case is north ascending. Then draw C M for the 
axis of the Moon's orbit, and bisect the angle M C H by 
the right line C Z. If the Moon's latitude had been 
north descending, the axis of her orbit would have been 
on the right hand from the axis of the ecliptic. 

The axis of the Moon's orbit lies the same way when 
her latitude is south ascending, as when it is north as- 
cending, and the same way when south descending, as 
when north descending. 

Take the Moon's latitude (40 minutes and 18 seconds) 
from the scale C A in your compasses, and set it from 
i to x in the bisecting line C Z, making % x parallel to 
C y and through x at right angles ; to the Moon's orbit 
(C M) draw the straight line N w x y s for the path of 
the penumbra's centre over the earth's disk. 

The point w in the axis of the Moon's orbit, is that, 
where the penumbra's centre approaches nearest to the 
centre of the earth's disk, and consequently is the middle 
of the general eclipse. The point x is where the con- 
junction of the Sun and Moon falls, according to equal 
time, as calculated by the tables, and the point y is the 
ecliptical conjunction of the Sun and Moon. 

Take the Moon's true horary motion from the Sun, 
(27 minutes and 54 seconds,) in your compasses, from 
the scale C A, (every division of which is a minute of a 
degree,) and with that extent make marks along the 



SEC. XVI.] DELINEATION OF SOLAR ECLIPSES. 189 

path of the penumbra's centre, and divide each space 
from mark to mark, into 80 equal parts, or horary min- 
utes, by dots, and set the hours to every 60th minute in 
such manner, that the dot signifying the instant of new- 
Moon by the tables, may fall into the point x, half way 
between the axis of the Moon's orbit and the axis of the 
ecliptic ; and then the remaining dots will be the points 
on the earth's disk, where the penumbra's centre is at 
the instants denoted by them in its transit over the earth. 

Apply one side of a square to the line of the penum- 
bra's path, and move the square backwards and forwards 
until the other side of it cuts the same hour and minute, 
(as at m and in) both in the path of the penumbra's cen- 
tre and the path of London ; and the particular minute or 
instant which the square cuts at the same time in both 
paths, will be the instant of the visible conjunction of the 
Sun and Moon, or the greatest obscuration of the Sun at 
the place for which the construction is made, (namely, 
London in this example,) and this instant is at 47 min- 
utes and 29 seconds past 10 o'clock in the morning, 
which is 17 minutes, 5 seconds later than the tabular 
time of true conjunction. 

Take the Sun's semidiameter, (16 minutes and six 
seconds,) in your compasses, from the scale C A, and 
setting one foot in the path of London at m, viz. at 47 
minutes and 30 seconds past 10, with the other foot de- 
scribe the circle U Y, which will represent the Sun's 
disk as seen from London at the greatest obscuration. 

Then take the Moon's semidiameter, 14 minutes and 
57 seconds, in your compasses, from the same scale, and 
setting one foot in the path of the penumbra's centre at 
m in the 47£ minutes after 10, with the other foot de- 
scribe the circle T Y for the Moon's disk, as seen from 
London at the time when the eclipse is at the greatest, 
and the portion of the Sun's disk, which is hidden or cut 
off by the disk of the Moon, will show the quantity of 
the eclipse at that time, which quantity may be mea- 
sured on a line equal to the Sun's diameter, and divide it 
into 12 equal parts for digits, which, in this example, is 
nearly 11 digits. This eclipse was annular at Paris, .• 



SEC. XVI.] DELINEATION OF SOLAR ECLIPSES. 191 

Lastly, take the semidiameter of the penumbra, 31 
minutes and 3 seconds, from the scale A C, in your com- 
passes, and setting one foot in the line of the penum- 
bra's path, on the left hand, from the axis of the ecliptic, 
direct the other foot towards the path of London, and 
carry that extent backwards and forwards, until both 
the points of the compasses fall into the same instants 
in both the paths, and these instants will denote the 
time when the eclipse begins at London. Proceed in 
the same manner on the right hand of the axis of the 
ecliptic, and where the points of the compasses fall into 
the same instants in both the paths, they will show at 
what time the eclipse ends at London. 

According to this construction, this eclipse began at 20 
minutes after 9 in the morning, at London, at the points 
N and O, 47 minutes and 30 seconds after 10, at the 
points rn and m for the time of the greatest obscuration, 
and 18 minutes after 12, at R and S, for the time when 
the eclipse ends. 

In this construction, it is supposed that the angles 
under which the Moon's disk is seen during the whole 
time of the eclipse, continues invariably the same, and 
that the Moon's motion is uniform and rectilinear dur- 
ing that time. But these suppositions do not exactly 
agree with the truth, and therefore supposing the ele- 
ments given by the tables to be accurate, yet the times 
and phases of the eclipse deduced from its construction, 
will not answer to exactly what passes in the heavens, but 
may be at least two or three minutes wrong, though the 
work may be done with the greatest care and attention. 

The paths also, of all places of considerable latitudes 
are nearer the centre of the earth's disk as seen from the 
Sun, than those constructions make them ; because the 
disk is projected as if the earth were a perfect sphere, 
although it is known to be a spheroid. The Moon's sha- 
dow will consequently go farther northward in all places 
of northern latitude, and farther southward in all places of 
southern latitude, than can be shown by any projection. 



192 DELINEATION OP LUNAR ECLIPSES. [sEC. XVII. 



SECTION SEVENTEENTH. 



THE PROJECTION OF LUNAR ECLIPSES. 

When the Moon is within 12 degrees of either of her 
nodes, at the time when she is full, she will be eclipsed, 
otherwise not, as before stated. 

Required the true time of full Moon at London, in 
May, 1762, New Style, and also whether there were an 
eclipse of the Moon at that time or not. 

It will be found by the precepts, that at the true time 
of full Moon in May, 1762, the Sun's mean distance from 
the ascending node was only 4 degrees, 49 minutes and 
36 seconds, and the Moon being then opposite to the 
Sun, must have been just as near her descending node, 
and was therefore eclipsed. The elements for the con- 
struction of lunar eclipses are eight in number, as fol- 
lows : — - 

1st. The true time of full Moon. 

2d. The Moon's horizontal parallax. 

3d. The Sun's semidiameter. 

4th. The Moon's semidiameter. 

5th. The semidiameter of the earth's shadow at the 
Moon. 

6th. The Moon's latitude. 

7th. The angle of the Moon's visible path with the 
ecliptic. 

8th. The Moon's true horary motion from the Sun. 

To find the true time of full Moon, proceed as directed 
in the Precepts, and the true time of full Moon in May, 
1762j will be found on the 8th day, at 50 minutes, and 
50 seconds past 3 o'clock in the morning. 

To find the Moon's horizontal parallax, enter Table 



SEC. XVII.] DELINEATION OF LUNAR ECLIPSES. 193 

15th with the Moon's mean anomaly, (at the time of the 
above full Moon,) namely, 9 signs, 2 degrees, 42 min- 
utes, 42 seconds, and with it take out her horizontal 
parallax, which, by making the requisite proportions 
will be found to be 57 minutes and 23 seconds. 

To find the semi-diameters of the Sun and Moon, enter 
Table 15th, with their respective anomalies, the Sun's 
being 10 signs, 7 degrees, 27 minutes, 45 seconds, and 
the Moon's 9 signs, 2 degrees, 42 minutes, 42 seconds, 
(in this case,) and with these take out their respective 
semidiameters, the Sun's 15 minutes and 56 seconds, 
and the Moon's 15 minutes and 38 seconds. 

To find the semidiameter of the earth's shadow at the 
Moon, add the Sun's horizontal parallax, which is always 
9 seconds, to the Moon's, which in the present case is 57 
minutes and 23 seconds, the sum will be 57 minutes 
and 32 seconds ; from which subtract the Sun's 
semidiameter, 15 minutes and 56 seconds, and there 
will remain 41 minutes and 36 seconds for the semi- 
diameter of that part of the earth's shadow, which the 
Moon then passes through. 

To find the Moon's latitude. Find the Sun's true dis- 
tance from the Moon's ascending node, (as already 
taught,) in the first Example for finding the Sun's true 
place, at the true time of full Moon, and this distance 
increased by 6 signs, will be the Moon's true distance 
from the same node, and consequently the argument for 
finding her true latitude. 

The Sun's mean distance from the ascending node 
was at the true time of full Moon, signs, 4 degrees, 49 
minutes, 35 seconds ; but it appears by the Example that 
the true time thereof, was 6 hours, 33 minutes and 38 
seconds sooner than the mean time, and therefore we 
must subtract the Sun's motion from the node during 
this interval, from the above mean distance signs, 4 
degrees, 49 minutes and 35 seconds, in order to have his 
mean distance from the node at the time of true full 
Moon. Then, to this apply the equation of his mean 
distance from the node, found in Table 13th, by Ids 

17 



194 DELINEATION OF LUNAR ECLIPSES. [SEC. XVII. 

mean anomaly, 10 signs, 7 degrees, 27 minutes, 45 sec- 
onds; and lastly, add 6 signs, and the Moon's true dis- 
tance from the ascending node, will be found as follows : — 

s. d. m. s. 
Sun from node at mean time of full Moon, 4 49 35 

( 6 hours, .... 15 35 
His motion from the node in < 33 minutes, . . 1 26 

( 38 seconds, . . 2 

Subtract the sum, 17 S 

Remains his mean dist. at true full Moon, .. 4 32 32 
Equation of his mean distance, add, 1 38 

Sun's true distance from the node, 6 10 32 

To which add, 60 

Moon's true distance from the node, 6 6 10 32 

And it is the argument used to find her true latitude at 
that time. Therefore, with this argument, enter Table 
14th, making proportions between the latitudes belong- 
ing to the 6th and 7th degree of the argument for the 10 
minutes and 32 seconds, and it will give 32 minutes and 
21 seconds for the Moon's true latitude, which appears 
by the table to be south descending. 

To find the angle of the Moon's visible path with the 
ecliptic. This may be always stated at 5 degrees and 
35 minutes without any error of consequence, in the 
projection of either solar or lunar eclipses. 

To find the Moon's true horary motion from the Sun, 
With their respective anomalies, take out their horary 
motions from Table 15th; and the Sun's horary motion, 
subtracted from the Moon's, leaves remaining the Moon's 
true horary motion from the Sun, in the present case, 30 
minutes and 52 seconds. 

The above elements are collected for use. 

D. h. m. s. 

1st. The time of full Moon in May, 1762, S 3 50 50 

D. 

2d. Moon's horizontal parallax, 57 23 



SEC.XVII.J DELINEATION OP LUNAR ECLIPSES. 195 

D. M. S. 

3d. Suns semidiameter, 15 56 

4th. Moon's semidiameterj 15 3S 

5th. Semidiameter of earth's shadow at Moon, 41 36 
6th. Moon's true latitude south descending, 32 21 
7th. Angle of Moon's visible path with ecliptic, 5 35 
8th. Moon's true horary motion from Sun, 30 52 
These elements being found for the construction of 
the Moon's eclipse in May, 1762, proceed as follows : — 
Make a scale of any convenient length, as W X and 
divide it into 60 equal parts, each part standing for a 
minute of a degree. Draw the right line A C B for part 
of the ecliptic, and A perpendicular thereto for the 
southern part of its axis, (the Moon having south lati- 
tude.) 

Add the semidiameters of the Moon and earth's 
shadow together, which in this case, make 57 minutes 
and 14 seconds ; and take this from the scale in your 
compasses, and setting one foot in the point C as a cen- 
tre, with the other describe the semicircle S D B, in one 
point of which the Moon's centre will be at the beginning 
of the eclipse, and the other at the end. 

Take the semidiameter of the earth's shadow, (41 
minutes and 36 seconds,) in your compasses from the 
scale, and setting one foot in the centre C, with the other 
describe the semicircle K L M for the southern half of 
the earth's shadow, because the Moon's latitude is south 
in this eclipse. 

Make C D equal to the radius of a line of chords on 
the sector, and set off the angle of the Moon's visible 
path with the ecliptic, (5 degrees and 35 minutes,) from 
D to E, and draw the right line C F E for the southern 
half of the axis of the moon's orbit, lying to the right 
hand from the axis of the ecliptic C A, because the 
Moon's latitude is south descending in this eclipse. It 
would have been the same way on the other side of the 
ecliptic, if her latitude had been north descending, but, 
contrary in both cases, if her latitude had been either, 
north or south ascending. 



196 DELINEATION OF LUNAR ECLIPSES. [sEC. XVII, 

Bisect the angle A C E by the right line C g, in 
which the true equal time of opposition of the Sun and 
Moon falls, as found from the tables. 

Take the Moon's latitude, 32 minutes and 21 seconds, 
from the scale in your compasses, and set it from C to 
G in the line C G g, and through the point g ; at right 
angles to C F E, draw the right line, P H G F N, for 
the path of the Moon's centre. Then F shall be the 
point in the earth's shadow, where the Moon's centre is 
at the middle of the eclipse ; G, the point where her cen- 
tre is at the tabular time of her being full ; and H, the 
point where her centre is at the instant of her ecliptical 
opposition. 

Take the moon's horary motion from the sun, (30 
minutes and 52 seconds,) in your compasses from the 
scale W X, and with that extent make marks along the 
line of the Moon's path P G N ; then divide each space 
from mark to mark, into 60 equal parts, or horary min- 
utes, and set the hours to the proper dots in such man- 
ner, that the dots signifying the instant of full Moon, 
namely, 50 minutes and 50 seconds after 3 in the morn- 
ing, may be in the point G, where the line of the Moon's 
path enters the line that bisects the angle D C E. 

Take the Moon's semidiameter, 15 minutes and 38 
seconds, in your compasses from the scale, and with 
that extent, as a radius upon the points N F and P as 
centres, describe the circle Q, for the Moon at the begin- 
ning of the eclipse, when she touches the earth's sha- 
dow at Y : the circle R for the Moon at the middle, and 
the circle S for the Moon at the end of the eclipse, just 
leaving the earth's shadow at W. 

The point N denotes the instant when the eclipse be- 
gins, namely, at 15 minutes and 10 seconds after two in 
the morning ; the point F, the middle of the eclipse, at 
47 minutes and 45 seconds after three ; and the point P, 
the end of the eclipse, at 18 minutes after five : at the 
greatest obscuration, the Moon was 10 digits eclipsed. 

The Moon's diameter, (as well as the Sun's,) is sup- 
posed to be divided into 12 equal parts, (called digits,) 



SEC. XVII.] DELINEATION OP LUNAR ECLIPSES. 197 

and so many of these parts as are darkened by the earth's 
shadow, so many digits is the Moon eclipsed. All that 
the Moon is eclipsed above 12 digits, shows how far the 
shadow of the earth is over the body of the Moon, on 
that edge to which she is nearest, at the middle of the 
eclipse. 



RULES FOR FINDING THE TIME WHEN A LUNAR 
ECLIPSE BEGINS AND ENDS, AND ALSO THE DIGITS 
ECLIPSED ARITHMETICALLY. 

To find the parts deficient. 

Rule. — To the Moon's horizontal parallax add the 
Sun's horizontal parallax, and from the sum subtract 
the Sun's semidiameter, and the remainder is the semi- 
diameter of the earth's shadow. To the semidiameter 
of the earth's shadow add the Moon's semidiameter, 
and from the sum subtract the Moon's latitude, and the 
remainder is the parts deficient. 

To find the digits eclipsed. 

Rule. — As the Moon's semidiameter is to six digits, so 
are the parts deficient to the digits eclipsed. 

To find the scruples of incidence. 

Ryle. — Reduce the sum of the Moon and earth's sha- 
dow to seconds, and also the Moon's latitude — find their 
sum and difference — multiply the sum by the difference, 
and the square root of the product will be the scruples 
of incidence in seconds. 

To find the time of half duration. 

Rule. — As the hourly motion of the Moon from the 
Sun is to one hour, so are the scruples of incidence to 
the time of half the duration of the eclipse. To the time 
of the middle of the eclipse, found by the tables, add the 
time of half duration, and the sum will be the time when 
the eclipse ends. Subtract it, and the remainder will 
show the time it commences. 

To calculate the time of total darkness, in total 
eclipses of the Moon. 

Rule. — Subtract the Moon's semidiameter from that 
17* 



198 DELINEATION OF LUNAR ECLIPSES. [SEC. XVII. 

of the earth's shadow, reduce the remainder, and also 
the Moon's latitude to seconds, find their sum and differ- 
ence, multiply the sum by the difference, and the square 
root of the product will be the scruples of total dark- 
ness ; then as the hourly motion of the Moon from the 
Sun is to one hour, so are the scruples of total darkness 
to the time of its half duration. To the middle of the 
eclipse add the time of half duration, and the sum will 
be the time when total darkness ends. Subtract it, and 
the remainder will show the time that total darkness 
commences. 

It is difficult to observe exactly, either the beginning 
or ending of a lunar eclipse, even with a good telescope ; 
because the earth's shadow is so faint and ill-defined 
about the edges, that when the Moon is either just touch- 
ing or leaving it, the obscuration of her limb is scarcely 
sensible, and therefore the closest observers can hardly 
be certain to four or five seconds of time. 

But both the beginning and ending of solar eclipses 
are instantaneously visible ; for the moment that the edge 
of the Moon's disk touches the Sun's, his roundness ap- 
pears to be broken on that part, and the moment she 
leaves it, he appears to be round again. 

In Astronomy, eclipses of the Moon are of great use in 
ascertaining the periods of her motions, especially such 
eclipses as are observed to be alike in ail circumstances, 
and have long intervals of time between them. In Geo- 
graphy, the longitudes of places are found by eclipses. 
The eclipses of the Moon are more useful for this pur- 
pose than those of the Sun, because they are more fre- 
quently visible, and the same lunar eclipse is equally 
large at all places where it is seen. 

In Chronology, both solar and lunar eclipses serve to 
determine exactly the time of any past event ; for there 
are so many particulars observable in every eclipse with 
respect to its quantity, the places where it is perceivable, 
(if of the Sun,) and the time of the day or night, that it 
is impossible that there can be two solar eclipses in the 
course of many ages, which are alike in all circum- 



SEC. XVII.] DELINEATION OF SOLAR ECLIPSES, 



199 



Moon's Eclipse in May, 1762. 



:>£ » ® S | 




200 ON THE FIXED STARS. [SECT. XVIII. 



SECTION EIGHTEENTH. 



ON THE FIXED STARS. 

The Stars are said to be fixed, because they have 
been generally observed to keep at the same distances 
from each other ; their apparent diurnal revolutions 
being caused solely by the earth's turning on its axis. 
They appear of a sensible magnitude to the eye, because 
the retina is affected not only by the rays of light which 
are emitted directly from them, but by many thousands 
more, which, falling upon our eyelids and upon the aerial 
particles about us, are reflected into our eyes so. strongly as 
to excite vibrations not only in those points of the retina 
where the real images of the stars are formed, but also 
in other points of some distance round. This makes us 
imagine the stars to be much larger than they would 
appear if we saw them only by the few rays which 
come directly from them, so as to enter our eyes, with- 
out being intermixed with others. Any person may be 
sensible of this, by looking at a star of the first magni- 
tude, through a long, narrow tube, which, though it 
takes in as much of the sky as would hold a thousand 
such stars, yet scarcely renders that one visible. 

The more a telescope magnifies, the less is the aper- 
ture through which the star is seen ; and consequently 
the less number of rays it admits into the eye. The stars 
appear less in a telescope which magnifies two hundred 
times, than they do to the naked eye ; insomuch that 
they seem to be only indivisible points : it proves at once 
that the stars are at immense distances from us, and 
that they shine by their own proper light. If they shone 
by reflection, they would be as invisible without tele- 



SEC. XVIII.] ON THE FIXED STARS. 201 

scopes as the satellites of Jupiter. These satellites ap- 
pear larger when viewed with a good telescope, than 
any of the fixed stars. 

The number of stars discoverable in either hemisphere 
by the unaided sight, is not above a thousand. This at 
first may appear incredible, because they seem to be 
almost innumerable ; but the deception arises from our 
looking confusedly upon them, without reducing them 
to any order : look steadfastly upon a large portion of 
the sky, and count the number of stars in it, and you 
will be surprised to find them so few. Consider only 
how seldom the Moon passes between us and any star, 
(although there are as many about her path as in any 
other parts of the heavens,) and you will soon be con- 
vinced that the stars are much thinner sown than you 
expected. The British catalogue, which, besides the stars 
visible to the naked eye, includes a great number which 
cannot be seen without the assistance of a telescope, 
contains no more than three thousand in both hemi- 
spheres. 

As we have incomparably more light from the Moon 
than from all the stars together, it is the greatest absurd- 
ity to imagine that the stars were made for no other pur- 
pose than to cast a faint light upon the earth, especially 
since many more require the assistance of a good tele- 
scope to find them out, than are visible without that in- 
strument. Our Sun is surrounded by a system of plan- 
ets and comets, all of which would be invisible from the 
nearest fixed star. And from what we already know of 
the immense distance of the stars, the nearest may be 
computed at thirty-two billions of miles from us, which 
is farther than a cannon ball can fly in seven millions of 
years, though it proceeded with the same velocity as at 
its first discharge. Hence it is easy to prove, that the 
Sun, seen from such a distance, would appear no larger 
than a star of the first magnitude. From the foregoing 
observations, it is highly probable, that each star is the 
centre of a magnificent system of worlds, moving round 
it, though unseen by us, and are irradiated by its beams : 



202 ON THE FIXED STARS. [SEC. XVIII. 

especially as the doctrine of plurality of worlds is ration- 
al, and greatly manifests the power, wisdom, and good- 
ness of the great Creator. 

The stars, on account of their apparently various 
magnitudes, have been distributed into several classes 
or orders. Those which appear largest, are called stars 
of the first magnitude ; the next to them in lustre, stars 
of the second magnitude ; and so on to the sixth, which 
are the smallest that are visible to the unaided sight. 
This distribution having been made long before the in- 
vention of telescopes, the stars which cannot be seen 
without the assistance of these instruments, are distin- 
guished by the name of telescopic stars. 

The ancients divided the starry spheres into particu- 
lar constellations, or systems of stars, according as they 
lay near each other, so as to occupy those spaces which 
the figures of different sorts of animals or things would 
take up, if they were there delineated. And those stars 
which could not be brought into any particular constel- 
lation, were called unformed stars. 

The division into different constellations or asterisms, 
serves to distinguish them from each other ; so that any 
particular star may be readily found in the heavens, by 
means of a celestial globe, on which the constellations 
are so delineated as to put the most remarkable stars into 
such parts of the figures as are most easily distinguished. 
The number of ancient constellations is 46, and upon 
our present globes about 70. There is also a division of 
the heavens into three parts. First, the zodiac, signify- 
ing an animal, because most of the constellations in it, 
which are twelve in number, are the figures of animals, 
as Aries, the ram ; Taurus, the bull ; Gemini, the twins ; 
Cancer, the crab ; Leo, the lion ; Yirgo, the virgin ; Libra, 
the balance ; Scorpio, the scorpion; Sagitarius, the archer ; 
Capri corn us, the goat ; Aquarius, the water-bearer ; and 
Pisces, the fishes. The zodiac goes quite round the 
heavens ; it is about sixteen degrees broad, so that it 
takes in the orbits of the Moon, and of all the planets, 
(excepting that of Pallas, and the satellites of Herschel.) 



SEC. XVIII.] ON THE FIXED STARS. 203 

Along the middle of this zone, or belt, is the ecliptic, or 
circle which the earth describes annually, as seen from 
the Sun, and which the Sun appears to describe as seen 
from the earth. Second, all that region of the heavens 
which is on the north side of the zodiac, containing 21 
constellations ; and third, that region on the south side 
of the zodiac, containing 15 constellations. 

There is a remarkable track around the heavens, 
called the Galaxy, or Milky Way, from its peculiar white- 
ness. It was formerly thought to be owing to a vast 
number of very small stars, closely connected, and the 
observations of Dr. Herschel have fully confirmed the 
opinion. He therefore considers the Galaxy as a very 
extensive branching congeries of many millions of stars, 
which probably owes its origin to several remarkably 
large, as well as very closely scattered small stars, that 
may have drawn together the rest. 

ON GROUPS OF STARS. 

Groups of stars succeed to clustering stars in Dr. 
Herschel's arrangement. A group is a collection of 
stars, closely, and almost equally compressed, and of any 
figure or outline. There is no particular condensation 
of the stars to indicate the existence of a central force, 
and the groups are sufficiently separated from neighbour- 
ing stars to show that they form peculiar systems of 
their own. 

ON CLUSTERS OF STARS. 

Dr. Herschel regards clusters of stars as the most 
magnificent objects in the heavens. They differ from 
groups in their beautiful and artificial arrangement. 
Their form is generally round, and their condensation 
is such as to produce a mottled lustre, somewhat resem- 
bling a nucleus. The whole appearance of a cluster 
indicates the existence of a central force, residing either 
in a central bodyj or in the centre of gravity of the 
whole system. 



204 ON THE FIXED STARS. [SEC. XVIII. 

Nebulae are light spots in the heavens, sometimes 
denominated cloudy stars (Plate 7, Fig. 7) : some of them 
are found to be clusters of telescopic stars. The most 
noted nebula was discovered by Huygens, in 1656. 
It is between the two stars in the sword of Orion. In 
one part of it, (Plate 8th, Fig. 1st,) a bright spot upon a 
dark ground seems to be an opening into a brighter and 
more distant region. Nebulae were discovered by Dr. 
Hawley and others ; but to Dr. Herschel, says Enfield, 
are we indebted for catalogues of 2000 nebulae and 
clusters of stars which he himself has discovered : Dr. 
Brewster says 2500. 

"What an astonishing view of the works of creation 
is opened upon us by the night. With wonder and 
delight we greet the return of day." The beauty and 
even the sublimity of this world are lighted up to us by 
the splendour of the morning. But how surpassed are 
these by the infinite grandeur presented to our view by 
the nocturnal heavens. To the night we are indebted 
for the most exalted conceptions we are capable of form- 
ing of the immensity and sublimity of the works of the 
omnipotent Jehovah. 

We cannot contemplate the fixed stars without the 
most profound awe. How inconceivably great and 
powerful, wise and good, must be the Author and Gov- 
ernor of all these resplendent luminaries, in which we 
behold, not a solitary world, but a system of worlds kept 
in harmonious motion by the Sun ; not one Sun and one 
system only, but millions of suns and of systems, ranged 
in endless perspective, all revolving in harmonious or- 
der, and rendering endless tribute of praise to Him who 
said, ' Let there be light,' and there was light. 



INTERROGATIONS FOR SECTION EIGHTEENTH. 

What is a fixed star ! 

Why do they appear of sensible magnitude to the eye % 
Do the Stars appear larger when viewed through a 
telescope than viewed with the eye only ? 



SEC. XVIII.] OF THE FIXED STARS. 205 

What does it prove ? 

Which appear the largest, the satellites of Jupiter, or 
the stars, when viewed with a telescope 1 

About how many stars in a clear night can be seen by 
the naked eye 1 

How many in the British catalogue 1 

Are some of that number telescopic ? 

Would the planets and comets of the solar system be 
invisible from the nearest star ? 

At how many miles distant may we with propriety 
suppose the nearest fixed star ? 

How lon£ would a cannon-ball be in flying that dis- 
tance, supposing it should continue to move with the 
same velocity, as at its first discharge ? 

Of what size would the Sun probably appear from the 
nearest fixed star ? 

Is it not probable that every star is the centre of a 
magnificent system ? 

On what account have they been distributed into 
classes ? 

What are those called which appear largest ? 

What are constellations ? 

What is the use of dividing them into constellations ? 

How many constellations on the celestial globes ? 

What is the Zodiac ? 

How many constellations in the Zodiac ? 

What is the breadth of the Zodiac 1 

What the Galaxy, or Milky Way ? 

What is a group of stars ? 

What are clusters of stars ? 

What are Nebulae? 



IS 



206 OF THE GREGORIAN CALENDAR. [SEC. XVIII. 



SECTION NINETEENTH. 



An account of the Gregorian, or New Style, together 
with some chronological problems, for finding the 
epact, golden number, dominical letter, $*c. 

Pope Gregory XIII. made a reformation of the calen- 
dar. The Julian calendar, (or Old Style,) had before that 
time, been in general use all over Europe. The year, 
according to the Julian calendar, consists of 365 days and 
6 hours, which 6 hours being i part of a day, the common 
years consisted of 365 days : and every fourth year one 
day was added to the month of February, which made 
each of those years consist of 366 days, commonly called 
leap years. 

This computation (though near the truth) is more 
than the Solar year, by 11 minutes and 3 seconds, which, 
in 131 years, amounts to a whole day ; by which the 
vernal equinox was anticipated ten days from the time 
of the general council of Nice, held in the year 325 of 
the Christian era, to the time of Pope Gregory, who 
therefore caused ten days to be taken out of the month 
of October, 1582, to make the equinox fall on the 21st 
of March, as it did at the time of that council ; and to 
prevent the like variation for the future, he ordered that 
three days should be abated in every four hundred years, 
by reducing the leap year at the close of each century, 
for three successive centuries,* to common years, and 
retaining the leap year at the close of each fourth cen- 
tury only. This, at that time, was esteemed as exactly 
conformable to the true solar year. But since that time, 
the true solar year is found to consist of 365 days, 5 
hours, 48 minutes and 49 seconds, which in 50 centu- 
ries will make another day's variation. 



SEC. XVIII.] CHRONOLOGICAL PROBLEMS. 207 

Though the Gregorian Calendar, (or New Style,) had 
long been in use throughout the greatest part of Europe 
it did not take place in Great Britain and America, till 
the first of January, 1752 ; and in September following, 
the 11 days were adjusted by calling the third day of that 
month, the fourteenth, and continuing the rest in their 
order. 

CHRONOLOGICAL PROBLEMS. 

As there are three leap years to be abated in every 
four centuries, — to find which century is to be leap year 
and which not. 

Rule. — Cut off two cyphers from the given year and 
divide the remaining figures by 4, if nothing remain, the 
year will be leap year. - 

The year ^-J^ there being a remainder of 3, it will 
not be leap year. But the year ^ will. 

TO FIND THE DOMINICAL OR SUNDAY LETTER. 

Rule, — To the given year, add its fourth part, reject- 
ing remainders, divide the sum by 7, and if there be no 
remainder, A is a Sunday Letter ; but if any number 
remains, then the letter standing under that number, is 
the Dominical letter, and the day of the week on which 
the year commences. 

A leap year has two Dominical letters, the first of 
which commences the year, and continues to the 24th 
of February, and the other to the end of the year. 

EXAMPLE. 

12 3 4 5 6 7 Required the Dominical letters 

>; ^ for the year 1832. 

. ^ £rS £ £ 4)1832 

ttlH-^l 458 

1 § 1 § M S| Days in a week 7)2290 

0123456 647 — L 

AGFEDCB 
The year 1832 was leap year, and according to the 



208 CHRONOLOGICAL PROBLEMS, [SEC. XIX. 

work, the remainder being 1, the first Sunday letter was 
A, and G was the second, the year also commenced on 
Sunday. 

To find the Golden Number. 

Rule. — Add 1 to the given year, divide the sum by 
19, and the remainder will be the Golden Number ; if 
nothing remain, then 19 will be the number sought. 

Required the Golden Number for the year 1832. 
To the given year 1832 

Add \_ 

19)1833(96 
171 
123 
114 



Golden Number, 9 

To find the Epact. 

Rule. — Subtract 1 from the Golden Number, divide 
the remainder by 3; if 1 remain, add 10 to the dividend, 
the sum will be the Epact ; if nothing remain, the divi- 
dend is the Epact. 

Required the Epact for the year 1832. The Golden 
Number as found above, is 9, therefore, subtract 1, and 
the remainder is 8, divide 8 by 3, and the quotient is two, 
and 2 remains, multiply this remainder by 10, and the 
product is 20, to which add the dividend, and the sum is 
29, the Epact for 1832. 

To find the year of the Dionysian Period. 

Rule. — Add to the given year 457, divide the sum hy 
532, and the remainder will be the number required. 

Required the year of the Dionysian Period, for the 
year 1832. To the given year, 1832 
Add 457 

532)2289(4 
2128 



161=Dionysiati Period, 
To find the Julian Period. 

Rule. — Add 4713 to the given year, and the sum will 
be the Julian Period. 



SEC. XIX.] CHRONOLOGICAL PROBLEMS. 209 

Required the Julian Period for the year of the Chris- 
tian Era, 1832. 1832 
4713 



6545 year of the Julian period. 
* To find the Cycle of the Sun, Golden Number, and 
Indiction for any current year. 

Rule. — To the current year add 4713, divide the sum 
by 28, 19, and 15, respectively, and the several remain- 
ders will be the numbers required. If nothing remains, 
the divisors are the required numbers. 

Required the Cycle of the Sun, Golden Number, and 
Indiction for the year 1832. 

1832 1832 1832 
4713 4713 4713 



28)6545(233 19)6545(344 15)6545(436 
56 57 60 



94 84 54 
84 76 45 



105 85 95 
84 76 90 



21=CycleoftheSun 9=Golden Number 5= Indiction. 

To find on what day Easter will happen. 

It was ordered by the Nicene Council, that Easter 
Sunday, should be kept on the first Sunday after the first 
full Moon which happened upon or after the twenty- 

* A Cycle is a perpetual round, or circulation of the same parts of 
time of any sort. The Cycle of the Sun, is a revolution of 28 years, 
in which the days of the months return again to the same days of the 
week, the Sun's place to the same signs and degrees of the ecliptic, 
on the same months and days, so as not to differ one degree in a hun- 
dred years, and the leap years begin the same course over again, with 
respect to the days of the week, on which the days of the months fall. 

The Cycle of the Moon, (commonly called the Golden Number,) 
is a revolution of 19 years, in which the conjunctions, oppositions, 
and other aspects of the Moon, are within an hour and a half of being 
the same as they were on the same days of the months 19 years be- 
fore. The indiction is a revolution of 15 years, used only by the 
Romans, for indicating the times of certain payments made by the 
subjects of the Republic: It was established by Constantine, A. D.312. 

18* 



210 CHRONOLOGICAL PROBLEMS. [SEC. XIX. 

first day of March, the day on which they thought the 
Vernal Equinox happened ; though this was a mistake, 
for the vernal equinox that year fell on the 20th of March : 
but yet, the full Moon which fell on or next after the 
twenty-first of March, they called the Paschal full Moon ; 
and by the introduction of the Gregorian, or New Style, 
the equinox will now always happen on the twentieth, 
or twenty-first of March : and if the full Moon happen 
on a Sunday, Easter Day is to be the next Sunday after. 
Therefore, find the time of the next full Moon after the 
twenty-first of March, and the following Sunday is 
Easter. 



SEC, 



XIX.] 



PERPETUAL ALMANACK- 



211 



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212 



ASTRONOMICAL TABLES. [SEC. XIX. 



TABULAR VIEW OF THE SOLAR SYSTEM. 



Names of 


Mean diame- 


the 


ters in 


Planets, 


miles. 


The Sun 


883,246 


Mercury- 


3,224 


Venus 


7,687 


Earth 


7,912 


Moon 


2,180 


Mars 


4,189 


Vesta 


238 


Juno 


1,425 


Ceres 


163) 
1,024 ( 


Pallas 


80) 
20,99 $ 


Jupiter 


89,170 


Saturn 


79,042 


Herschel 


35,112 



Mean distances 

from the Sun in 

round numbers of 

miles 



37,000,000 

68,000,000 

95,000,000 

95,000,000 

144,000,000 

225,000,000 

252,000,000 

263,000,000 

265,000,000 

490,000,000 

900,000,000 

1,800,000,000 



The correct 


Mean appa- 


mean dis- 


rent diame- 


tances, that 


ters as seen 


of the earth 


from the 


being 100000 


earth. 




M. s. 3ds. 




32 1 30 


38,710 


10 


72,333 


58 


100,000 




100,000 


31 8 


152,369 


27 


237,300 


30 


265,700 


3 


276,500 


1 > 
6,4 C 


279,100 


,5 \ 

6,5 \ 


520,279 


39 


954,072 


18 


1,908,352 


3 54 



Mean appa 
rent diame- 
ter as seen 
from the 
sun. 



16seconds 
30 
17,2 
4,6 
10 



TABULAR VIEW OF THE SOLAR SYSTEM. 



Names of Tropical revolu- Sidereal revolu- Place of Aphe- 
lion in Jan. 1800. 



the Planets. 



tions. 



Sun 



M. 



I D. 



H . 



M. S. I D. H. M. 



Mercury- 
Venus 
Earth 

! Moon 

I Mars 

| Vesta 
Juno 
Ceres 
Pallas 
Jupiter 
Saturn 
Herschel 



87 23 14 32 
224 16 41 27 
365 5 48 49 



686 22 
1155 4 

1588 
1681 



18 2' 



4330 14 39 2 
10746 19 16 15 
30637 4 



87 23 

224 16 

365 6 

686 23 



1703 16 

4332 14 

10759 1 

30737 18 



15 


34 


49 


10 





12 


30 


35 


48 




27 


10 


51 


11 









8 14 20 50 
10 7 59 1 

9 8 40 12 



2 24 4 

9 42 53 

29 49 33 

25 57 



15 

7 

8 20 



8 29 4 11 
11 16 30 31| 



SEC. XIX, 



ASTRONOMICAL TABLES. 



213 



TABULAR VIEW OE THE SOLAR SYSTEM. 



1 Names of 


j Densi- 1 Diurnal rotation 


Propor- 


Inclinations 


Inclinations 1 


the planets. 


Ities, that around their 


Dwn 


tional 


of axes to 


of orbits to | 


1 


lof water 


axes. 




quanti- 


their orbits. 


the ecliptic in 




being 1. 






ties of 
matter. 




1780. 




! Id. 


H. M. 


S. 1 1 D. M. 1 D. M. S. 


The Sun 


1,1333 


25 


14 8 





33,3928 


82 44 




Mercury 


9,1666 


14 


24 5 


28 


0,1654 


unknown. 


7 


Venus 


5,7333 





23 20 


54 


0,8899 


75 


3 23 35 


Earth 


4,5 


1 








h 


66 32 




mean rate. 


Moon 




27 


7 43 


12 


0,025 


88 17 


5 9 3 


Mars 


3,2857 





24 39 


22 


0,0875 


59 22 


1 51 


Vesta 














7 8 46 


Juno 






27probably 






21 


Ceres 


2 















10 37 


Pallas 


2 















34 50 40 


Jupiter 


1,04166 





9 55 


37 


312,1 


90 nearly. 


1 18 56 


Saturn 


,406 





10 16 


2 


97,76 


60 probably. 


2 29 50 


Herschel 


,99 


unknown. 




16,84 




46 20 



TABULAR VIEW OF THE SOLAR SYSTEM. 



Names of the 
planets. 


Motion of 

aphelion 

in 100 

years. 


Longitude of 

ascending node 

in 1750. 


Motion of 1 Eccentri- 
nodes in cities, the 
10 years, mean dis- 
tances be- 
linglOOOOO. 


Greatest 

equation of 

centre. 


Sun 


d. m. s. Is. d. m. s. Id. m. s.l Id. m. s. 


Mercury 

Venus 

Earth 

Moon 

Mars 

Vesta 

Juno 

Ceres 

Pallas 
Jupiter 
Saturn 
Herschel 


1 33 45 
1 21 

19 35 

1 51 40 

1 34 33 

1 50 7 
1 29 2 


1 15 20 43 

2 14 26 18 

1 17 38 38 

3 13 1 
in 1804 

4 21 6 

2 20 58 401 
in 1802 

2 21 6 Of 
in 1804 J 

5 22 28 57 

2 7 55 32 

3 21 32 22 
2 12 47 


1 12 10 
51 40 

46 40 


7955,4 
498 
1681,39 

14183,7 
9322 

25096 

8141 

24630 
25013,3 

53640,42 

90804 


23 40 

47 20 

1 55 30 

10 40 40 

9 20 8 

28 25 

5 30 38 

6 26 42 

5 27 16 


59 30 

55 30 

1 44 35 



214 PROBLEMS. [SEC. XX. 



SECTION TWENTIETH. 



PROBLEMS TO BE SOLVED BY THE TERRESTRIAL GLOBE. 

A few problems are here inserted for such pupils as 
have the privilege of globes, and are under the care of 
instructers, who can explain the use of the different cir- 
cles and appurtenances belonging to them. 

problem i. 

To find the latitude of any given place. 

Bring the place of the graduated side of the brazen 
meridian, and the degree of the meridian over the place 
is the latitude required. 

What is the latitude of Boston ? 

Ans. 42° 28' north. 

Find the latitude of New York, Philadelphia, Wash- 
ington, New Orleans, Quebec, Amsterdam, London, 
Paris, Moscow, Constantinople, Rome, Detroit, Alexan- 
dria, Vienna, Lima, Stockholm, Tripoli, Savannah, 
Utica, Charleston, Naples, Warsaw, Cape Horn, Cairo, 
Cape Farewell, Canton. 

problem II. 
To find the longitude of any given place. 
Bring the place of the brazen meridian, and the degree 
of the equator under the meridian is the longitude. 
What is the longitude of Washington from London 7 

Ans. 76° 56' west. 
What is the longitude of Petersburgh ? 

Ans. 30° 15' east. 
What is the longitude of Philadelphia from Washing- 
ton ? Ans. 1° 56' east. 



SEC. XX.] PROBLEMS. 215 

Find the longitude from Wasnington of the following 
places : — 

New- York, Buffalo, Detroit, Boston, Quebec, Paris, 
Rome, Naples, Canton, Cairo, Warsaw, London, Vienna, 
Constantinople, Amsterdam. 

PROBLEM III. 

To find the difference of latitude of any two places. 

Find the latitude of each place by Problem I. If both 
are north or both south latitude, subtract the less from 
the greater ; but if one be north and the other south, add 
them together, the result will be the difference of 
latitude. 

1. What is the difference of latitude between Phila- 
delphia and Petersburgh ? Ans. 20°. 

2. What is the difference of latitude between Boston 
and Cape Horn 7 Ans. 97° 30'. 

The difference of latitude is required between Wash- 
ington and New York, between New York and London, 
between Albany and Gibraltar, between Boston and Con- 
stantinople, between Paris and Pekin, between Phila- 
delphia and Calcutta, between Boston and Baltimore, 
between Edinburgh and New Orleans. 

PROBLEM IV. 

To find the difference of longitude between any two 
places. 

Find, by Problem II, the longitude of each place. If 
both be in east or both in west longitude, subtract the 
less from the greater, and the result will be the differ- 
ence : but if one be east and the other west, add them 
together, and the sum, if less than 180 degrees, will be 
the difference required ; but if more, subtract the sum 
from 260, the remainder is the difference required. 

What is the difference of longitude between New 
York and Philadelphia, between Albany and Charleston, 
between Paris and New Orleans, between Pekin and 
Boston, between Hartford and Detroit, between Cairo 
and Petersburgh, between the mouth of the Mississippi 



216 PROBLEMS. [SEC. XX, 

and that of Columbia river, between Utica and Ro- 
chester. 

problem v. 

To find the distance in miles between any two places 
on the globe. 

Lay the quadrant of altitude over both places, and it 
will show the number of degrees between ; which mul- 
tiply by 69£, and the product will be the distance. 

1. What is the distance between London and Jamaica? 

Ans. 671°, equal to 4691,25 miles. 

2. What is the distance between Washington and 
Petersburgh, between New York and New Orleans, 
between London and Quebec, between Paris and Phila- 
delphia, between Canton and Charleston, between Cape 
Horn and Good Hope, between St. Helena and Boston, 
between London and Moscow. 

problem vl 

The hour of the day at any place being given, to find 
at what hour it is at any other place. 

Bring the place where the hour is given to the brazen 
meridian ; set the index to the given hour, then turn the 
globe till the proposed place comes under the meridian, 
the index will point to the hour required.* 

1. When it is eight o'clock A. M. at Boston, what is 
the time at Cape Farewell. Ans. 10 A. M. 

2. When it is 12 o'clock at noon at London, what is 
the time at Washington ? 

Ans. 6 hours, 52 minutes, 16 seconds, A. M. 

When it is midnight at New York, what hour is it at 
Paris, at Canton, at New Orleans, at Rome, at Peters- 
burgh, at Detroit, at Naples, at Warsaw ? 

When it is noon at Philadelphia, what is the hour at 
Boston, at Quebec, at Cairo, at Constantinople, at Jeru- 
salem, at Mexico, at St. Helena, at Moscow, at Alexan- 
dria, at Yienna, at Naples ? 

*If the required be west of the given place, turn the globe eastward ; 
but if east, turn the globe west. 



^SEC. XX. PROBLEMS. 217 

PROBLEM VII. 

The hour of the day being given at any place, to find 
all the places on the globe where it is at any other 
given hour. 

Bring the place to the brazen meridian, and set the 
index to the hour of that place. Turn the globe till the 
index points to the other given hour ; then all the places 
under the meridian are the places required. 

1. When it is 12 o'clock at noon in London, at what 
places is it 8 in the morning ? 

Ans. Cape Canso, Martinico, Trinidad, &c. 

2. When it is 2 o'clock P. M. at London, where is it 
halfpast5P. M.? 

Ans. Caspian Sea, Madagascar, Socotra, <fec. 

3. When it is 4 o'clock A. M. at New York, where is 
it noon ? at what places midnight ? 

4. When it is noon at Washington, where is it 9 
o'clock at night ? where midnight ? 

5. When it is midnight at Paris, where is it 8 o'clock 
in the morning ? where 8 at night ? 

6. When it is 6 o'clock at Jerusalem, where is it noon ? 
at what places is it midnight ? at what places 5 o'clock 
P.M.? 

7. When the sun is on the meridian at Alexandria, at 
what places is it midnight ? at what places 9 o'clock A. 
M. ? at what places 5 P. M. ? at what places 5 A. M. ? 

PROBLEM VIII. 

To find the antipodes of any place. 

Bring the given place to the meridian, and find its 
latitude. Set the index to 12, and turn the globe till the 
index points to the other 12 ; then the same degree of 
latitude on the other side of the equator shows the anti- 
podes 1 

1. What is the antipodes of London ? 

Ans. The south part of New Zealand. 

2. What is the antipodes of the Bermudas ? 

Ans. The south west part of New Holland. 
19 



218 PROBLEMS. [SEC. XX. 

3. What is the antipodes of the Society Islands ? 

Ans. The Red Sea. 

4. What is the antipodes of New- York ? Of London, 
of Naples, of the Caspian Sea, of Egypt, of Moscow, of 
St. Helena, of Spain, of Canton, of (Quebec 1 

PROBLEM IX. 

To find at what rate per hour the inhabitants of any 
given place are carried by the revolution of the earth on 
its own axis. 

Find how many miles make a degree of longitude in 
the latitude of the given place from the following table ; 
which multiply by 15, the product will be the number of 
miles per hour.* 



TABLE, 

Showing the length of a degree of longitude for every 
degree of latitude, in geographical miles. 



«lat. 


miles. 


°lat. 


miles. 


°lat. 


miles. 


°lat. 


miles. 


1 


59,96 


24 


54,81 


47 


41,00 


69 


21,51 


2 


59,94 


25 


54,38 


48 


40,15 


70 


20,52 


3 


59,92 


26 


54,00 


49 


39,36 


71 


19,54 


4 


59,86 


27 


53,44 


50 


38,57 


72 


18,55 


5 


59,77 


28 


53,00 


51 


37,73 


73 


17,54 


6 


59,67 


29 


52,48 


52 


37,00 


74 


16,53 


7 


59,56 


30 


51,96 


53 


36,18 


75 


15,52 


S 


59,40 


31 


51,43 


54 


35,26 


76 


14,51 


9 


59,20 


32 


50,88 


55 


34,41 


77 


13,50 


10 


58,18 


33 


50,32 


56 


33,55 


78 


12,48 


11 


58,89 


34 


49,74 


57 


32,67 


79 


11,45 


12 


58,68 


35 


49,15 


58 


31,70 


80 


10,42 


13 


58,46 


36 


48,54 


59 


30,90 


81 


09,38 


14 


58,22 


37 


47,92 


60 


30,00 


82 


08,35 


15 


58,00 


38 


47,28 


61 


29,04 


83 


07,32 


16 


57,60 


39 


46,62 


62 


28,17 


84 


06,28 


17 


57,30 


40 


46,00 


63 


27,24 


85 


05,23 


18 


57,04 


41 


45,28 


64 


26,30 


86 


04,18 


19 


56,73 


42 


44,95 


65 


25,36 


87 


03,14 


20 


56,38 


43 


43,88 


m 


24,41 


88 


02,09 


21 


56,00 


44 


43,16 


67 


23,45 


89 


01,05 


22 


55,63 


45 


42,43 


68 


22,48 


90 


00,00 


23 


55,23 


46 


41,68 











* The Sun passes one degree in four minutes, exactly, which is 
equal to fifteen degrees in one hour. 



SEC. XX.] PROBLEMS. 219 

At what rate per hour are the inhabitants of the fol- 
lowingplaces carried by the motion of the earth onits axis? 

Washington, New York, Quebec, Calcutta, Canton, 
Petersburgh, Cape of Good Hope, London, Quito, New 
Orleans. 

PROBLEM X. 

The day of the month being given, to find the Sun's 
place or longitude in the ecliptic, and its declination. 

Look for the given day in the circle of months on the 
horizon, and opposite to it in the circle of signs are the 
sign and degree in the ecliptic, and it will be the Sun's 
place of longitude. Bring this place to the meridian, 
and you will have the declination. 

1 . What is the Sun's longitude and declination on the 
22d of February ? 

Ans. 337° 30' from Aries, that is in Pisces 4° 30' : its 
declination is 10° south. 

2. What is the Sun's longitude and declination on 
the 5th of April? 

Ans. 25 degrees and 30 minutes in Aries ; its decli- 
nation 10° north. 

3. When does the Sun enter each of the signs ? 

4. What is the Sun's place and declination on the 
22d of December, on the 21st of June, on the 1st of 
September, on the 1st of January, on the 4th of July, 
and the 17th of October? 

PROBLEM XI. 

To rectify the globe for the latitude, zenith and Sun s 
place on any day. 

For the latitude. Elevate the pole till the horizon 
cuts the brass meridian in the degree corresponding to 
the latitude; the given place is then in the zenith. 
Then by problem tenth, find the Sun's place for the 
given day ; bring it to the meridian and set the index 
to 12.* 

* If the given place be in north latitude, elevate the north pole ; if 
3n south latitude, elevate the south pole. 



220 PROBLEMS. [SEC. XX 

1. Rectify the globe for the latitude of London on the 
10th of May. In this case elevate the north pole 51 
degrees and 30 minutes, then will London be in the 
zenith over it ; screw the quadrant of altitude : the 10th 
of May on the horizon, answers to the twentieth degree 
of Taurus, which find on the ecliptic, and bring it to the 
meridian, and set the index to 12. This is the position 
of the globe as it appears to the inhabitants of London 
on the 10th of May. 

2. Rectify the globe for New York January 21st, 
for Boston April 6th, for Washington June 16th, for 
Constantinople December 12th, for Petersburgh on the 
12th of October, for Jerusalem on the 3d day of April, 
for New Orleans on the 8th of January, and for Paris on 
the 22d day of February. 

PROBLEM XII. 

The month and day of the month being given, to 
find all those places on the globe which will have a ver- 
tical Sun on that day. Find the Sun's place in the 
ecliptic (by problem 10th,) and bring it to the meridian, 
turn the globe round, and all the places that pass under 
that degree of the meridian, will have a vertical Sun on 
that day. 

1. Find all the places which have a vertical Sun on 
the 22d day of February. 

Ans. Amazonia, Angola, Q,ueen Charlotte's Island, 
New Guinea and Peru. 

2. What places have a vertical Sun on the 6th day of 
June, on the 4th of July, on the 21st of September, on 
the 17th of October, and 12th day of March 1 

PROBLEM XIII. 

To find the time of the rising and setting of the Sun 
at any place, at any given day in the year, and the 
length of the day and night at that place. 

Rectify the globe (by problem 11th,) for the latitude of 
the place, find the Sun's place in the ecliptic (by pro- 
blem 10th.) and bring it tp the meridian, and set the index 



SEC. XX.] PROBLEMS. 221 

to 12. Bring the Sun's place to the eastern edge of the 
horizon, and the index will show the time of the Sun's 
rising. Bring it to the western edge of the horizon, and 
the index will show the time of the Sun's setting. 
Double the time of the Sun's rising and it will give the 
length of the night. Double the time of setting, and it 
will show the length of the day. 

At what time does the Sun set and rise at New York 
on the 10th day of May, and what is the length of the 
day and night ? 

Ans. It rises 56 minutes past four, and sets 4 minutes 
after seven. Length of the day 14 hours and 8 min- 
utes ; and length of the night 9 hours and 52 minutes. 

What is the time of the Sun's rising and setting, and 
the length of the day and night at Boston on the 8th of 
September, at Washington on the tOth of June, at 
London on the 22d of December, at Petersburgh on the 
21st of June, at Paris on the 1st of March, and at the 
Island of St. Helena on the 24th of October ? 

PROBLEM XIV. 

To find the length of the longest and shortest days 
and nights in any part of the earth. 

If the place be in the northern hemisphere, rectify the 
globe for the latitude of the place, bring the first degree 
of Cancer to the meridian, and proceed as in the last 
problem. If the place be in the southern hemisphere 
bring the first degree of Capricorn to the meridian, and 
proceed as before. 

1. What is the length of the longest day and short- 
est night at New York. 

Ans. Longest day 14 hours and 56 minutes. Short- 
est night 9 hours and 4 minutes. 

2. What is the length of the longest day and shortest 
night at each of the following places : — Boston, Wash- 
ington, Albany, Philadelphia, Mexico, Paris, London, Pe- 
tersburgh, Stockholm, Quebec, Jerusalem, Iceland, Can- 
ton, Berlin, Warsaw, Constantinople, St. Helena, Cal- 
cutta, and Van Dieman's Land. 

19* 



222 PROBLEMS. [SEC. XX. 

PROBLEM XV. 

The month and day of the month being given, to find 
those places where the Sun does not set, and where it 
does not rise on the given day. 

Find the Sun's declination, (by problem 10th,) elevate 
the pole for the declination in the same manner as for 
the latitude, turn the globe on its axis, and on the places 
around the pole, above the horizon, the Sun does not set ; 
and on the places around the other pole, below the ho- 
rizon, the sun does not rise on that day. 

1. How much of the south frigid zone is in darkness, 
and how much of the north frigid zone is enlightened 
on the 20th of May? 

Ans. 20 degrees round each pole. 

2. On which .pole does the Sunrise on the 6th of 
November ? 

3. Which frigid zone, and how much of it, has con- 
stant night on July 4th ? 

4. What days in the year does the Sun shine equally 
on both poles ? 

5. How much of the north frigid zone has constant 
day on the following days, namely, March 25th, April 
10th, June 21st, July 4th, August 8th, September 12th ? 



PROBLEMS— TO BE SOLVED BY THE CELESTIAL 
GLOBE. 

PROBLEM XVI. 

To find the right ascension of the Sun, or a star. 

Bring the Sun's place in the ecliptic, or the star, to the 
brass meridian ; then the degrees of the equinoctial under 
the meridian, reckoning from Aries eastward, is the right 
ascension. 

1. What is the Sun's right ascension on the 19th of 
April ? Ans. 27 degrees and 30 minutes. 

2. What is the Sun's right ascension on the 1st day of 
December 7 Ans. 247 degrees and 50 minutes. 

3. What is the Sun's right ascension on the 7th of 



SEC. XX.] PROBLEMS. 223 

January, 12th of February, 10th of March, 16th of April, 
21st of May, 18th of June, 31st of August, and 10th of 
September ? 

4. What is the right ascension of Aldebaran ? 

Ans. 66 degrees 6 minutes. 

5. What is the right ascension of Alioth, Arcturus, 
Bellatrix, Castor, Algol, Fomalhaut, Hyades, Pleiades, 
Procyon, Regulus, Rigel, Sirius, Antares, andPollux 1 

PROBLEM XVII. 

To find the declination of the Sun, or a star. 

Bring the Sun's place in the ecliptic, or the star, to 
the brass meridian, and the degree of the meridian over 
that place will be the declination. 

1. What is the Sun's declination on the 19th day of 
April ? Ans. 11 degrees 19 minutes. 

2. What is the Sun's declination January 12th, Febru- 
ary 20th, May 22d, September 7th, October 12th, and 
December 1st? 

3. What is the declination of Aldebaran ? 

Ans. 16 degrees 6 minutes. 

4. What is the declination of Mair, Arcturus, Algenib, 
Procyon, Regulus, Regel, Sirius, Antares, Pollux ? 

PROBLEM XVIII. 

The latitude of the place, and day and hour being 
given, to place the globe so as to represent the appearance 
of the heavens at that time at the place, and to point out 
the situations of the several stars. 

Elevate the pole for the latitude of the place, find the 
Sun's place in the ecliptic, and bring it to the meridian, 
and set the index to 12 ; if the time be afternoon turn the 
globe westward, if in the forenoon eastward, till the in- 
dex points to the given hour. The surface of the globe 
then represents the appearance of the heavens at that 
place at the given hour. 

1. Represent the appearance of the heavens for Janu- 
ary 1st, at 4 in the morning, and 9 at night ; April 21st, 



224 PROBLEMS. [SEC. XX. 

at 1 1 at night ; May 9th, at 2 in the morning ; June 16th, 
at midnight ; and November 12th, at midnight. 

PROBLEM XIX. 

To find the latitude or longitude of a given star. 

Screw the quadrant on the pole of the ecliptic, bring 
the star to the meridian, and the degrees of the quadrant 
between the ecliptic and star show the latitude ; and the 
degree of the ecliptic under the graduated edge of the 
quadrant, is the longitude. 

1. What is the latitude and longitude of Arcturus? 

Ans. Lat. 31 degrees north, long. 201 degrees. 

2. What are the latitudes and longitudes of Canis Mi- 
nor, Canis Major, Fomalhaut, Regulus, Sirius, Procyon, 
Alioth, Gamma, Castor, Antares, and Pollux ? 

PROBLEM xx. 

To find the distance in degrees between any two given 
stars on the celestial globe. 

Lay the quadrant of altitude over the two given stars, 
and the number of degrees between them, as reckoned 
on the quadrant, will be their distance as seen from the 
earth. Or extend a thread over any two given stars, 
apply the distance found to the equator, and count the 
number of degrees. 

1. What is the distance, in degrees, between Altair in 
the Eagle, and Sega in Lyra ? 

.2. Between Pollux in Gemini and Altair ? 3. Between 
Spica and Regulus? 4. Between Castor and Pollux? 
5. Between Rigel and Aldebaran ? 6. Between Sirius 
and Procyon? 7. Between Arcturus and Procyon? S. Be- 
tween Gamma and Alioth? 9. Between Sirius and 
Deneb ? 10. Between Regel and Sirius ? 



DICTIONARY 



ASTRONOMICAL TERMS 



Aberration, is an apparent motion of the celestial 
bodies, arising from the progressive motion of light and 
the earth's annual motion in its orbit. 

Acceleration of a Planet. A Planet is said to be accele- 
rated when its real diurnal motion exceeds its mean 
diurnal motion. 

Acceleration of the Moon, is a term used to express 
the increase of the Moon's mean motion from the Sun. 

Aldebaran, a star of the first magnitude in the sign 
Taurus. This star frequently suffers an occultation 
by the Moon, when the ascending node is in Virgo. 

Alioth, a star of the third magnitude in the tail of 
Ursa Major. 

Altitude of a Celestial Body, is the arc of a vertical 
circle measured from the horizon. 

Amphiscians, are the people who inhabit the torrid 
zone. 

Amplitude, is an arc of the horizon intercepted be- 
tween the east or west point, and the centre of the Sun 
or star at its rising or setting. 

Andromeda, a northern constellation, containing, ac- 
cording to Flamstead, sixty-six stars. 

Angle, is the inclination of two lines or planes, meet- 
ing in a point, and may be any number of degrees less 
than 180. 

Angle of Commutation, is the angle at the Sun, form- 
ed by two lines, one drawn from the earth, and the other 



226 • ASTRONOMICAL TERMS. 

from the place of the planet reduced to the ecliptic, meet- 
ing in the Sun's centre. 

Angle of Elongation, is the angle formed by two lines 
drawn from the earth, the one to the Sun and the other 
to the planet ; or it is the difference between the Sun's 
place, and the geocentric place of the planet. 

Angular motion, is the motion of the planets about the 
centre of the Sun, or it is that of the satellites about the 
centres of their primaries. 

Annual equation, is the difference between the planet's 
mean and true place. 

Anomalistic year, is the time of the Sun's leaving its 
apogee till it returns to it again, which is 365 days, 6 
hours and 14 minutes. 

Anomaly, is the distance of a planet, in degrees, min- 
utes and seconds, from the aphelion or apogee. 

Antarctic Circle, is a small circle parallel to the equa- 
tor, 23 degrees and 28 minutes from the south pole. 

Antares, the Scorpion's heart, a star of the first mag- 
nitude in the Constellation Scorpio. 

Antecedentia, a term made use of to signify that a 
planet moves retrograde, or contrary to the order of the 
signs, that is, from east to west. 

Antipodes, are the people of two places diametrically 
opposite to each other ; they differ in longitude 180 de- 
grees, the one having the same degree of north latitude 
that the other has of south. 

Aphelion, is that point in the orbit of the earth or 
planet which is at the greatest distance from the Sun. 

Apogee, is that point in the Moon's orbit which is 
farthest from the earth. 

Apses or Apsides, are the two points in the orbits of 
the planets or satellites, which are at the greatest and 
least distance from the centre of motion, and a line join- 
ing these two points is called the line of the Apsides. 

Aquarius, one of the constellations of the Zodiac, con- 
taining one hundred and eight stars. 

Arctic Circle, a small circle surrounding the north 
pole, and distant from it 23 degrees and 28 minutes. 



ASTRONOMICAL TERMS. 227 

Arcturus, a star of the first magnitude in the constel- 
lation Bootes. 

Aries, one of the northern constellations of the Zo- 
diac ; it contains 66 stars. 

Ascending latitude, is the latitude of the Moon or 
planet when going northward. 

Ascending node, is that point of the Moon's or planet's 
orbit, where it cuts the ecliptic in going northward. 

Aspect, is the situation of the stars or planets with 
respect to each other. 

Astronomy, from Aster a star, and Nomos a law, is 
the science by which we are taught the motion, magni- 
tudes and distances, &c, of the heavenly bodies. 

Atmosphere, is that elastic invisible fluid, which sur- 
rounds the globe and causes the refraction and twilight. 

Attraction, is that innate principle of matter by which 
bodies mutually tend towards each other. 

Aurora, the morning twilight. 

Aurora Borealis, or Northern Lights, a kind of meteor 
of pale colour, sometimes seen in the northern parts of 
the heavens, supposed to be an electrical phenomenon. 

Autumn, the third quarter of the year, which begins 
when the sun enters Libra, which is about the 21st or 
22d of September. 

Azimuth, of the celestial bodies, is an arc of the hori- 
zon, intercepted between the meridian and a vertical cir- 
cle passing through the body. 

Azimuth Compass, an instrument for finding the mag- 
netic azimuth, or amplitude of a celestial body. 

B. 

Barometer, an instrument for showing the gravity of 
the atmosphere. It is also useful for the ascertaining 
the altitudes of mountains, and correcting the variation 
of the refraction arising from the changes in the density 
of the atmosphere. 

Beard of a Comet, are the rays which it emits from its 
head in the direction of its motion. 

Bellatrix, a star of the second magnitude, in the left 
shoulder of Orion. 



228 ASTRONOMICAL TERMS. 

Bissextile, or leap year, is a year consisting of 366 
days, which happens every fourth year. 

Bootes, a northern constellation, containing fifty-four 
stars. 

Boreal Signs, are those on the north side of the equa- 
tor, viz. Aries, Taurus, Gemini, Cancer, Leo and Yirgo. 

C. 

Cancer, the Crab, one of the constellations of the 
Zodiac : when the Sun enters this sign it has its great- 
est north declination ; it contains 83 stars. 

Canis Major, or the Great Dog, is a southern constel- 
lation that contains Sirius, one of the brightest fixed 
stars in the heavens : the number of stars in this con- 
stellation is thirty-one. 

Cardinal points, are the north, south, east and west 
points of the horizon. 

Cassiopeia, a northern constellation, consisting of fifty- 
five stars. 

Centrifugal Force, is that force by which all bodies 
moving about a central body or force, endeavour to fly 
off in tangent lines. 

Centripetal Force, is that by which a body moving 
round another, tends towards it ; this, and the centrifugal 
force acting upon the planets, cause them to describe 
curvilinear orbits about the sun. 

Ceres, a primary planet, moving between the orbits of 
Mars and Jupiter. It was discovered on the first of 
January, 1801, by M. Piazzi. 

Conjunction of two celestial bodies, is when they have 
the same degree of longitude. 

Consequentia, in astronomy, is when the planets move 
according to the order of the signs. 

Constellation, is a number of stars contained within 
some assumed figure. 

Culmination, the transit or passage of a star over the 
meridian. 

Cycle, a certain period of time in which the same rev- 
olutions begin again. 



ASTRONOMICAL TERMS. 229 

Cycle of the Moon, or the Lunar Cycle, is a period of 
19 years, in which the new and full Moons return on 
the same days as they did nineteen years before : this 
Cycle is called the Golden Number. 

Cycle of the Sun, is a period of 28 years, in which 
time the days of the month return again to the same 
days of the week. 

D. 

Day Astronomical. The astronomical day begins at 
apparent noon, and is counted twenty-four hours to the 
following noon. 

Declination, is the distance of the Sun, Moon or stars 
from the equinoctial, either north or south. 

Degree, the thirtieth part of a sign. 

Descending Node, is that point of a planet's orbit where 
it cuts the ecliptic proceeding southward. 

Dial, an instrument to show the hour of the day by the 
Sun- 
Digit, the twelfth part of the Sun's or Moon's diame- 
ter, and is used to show the degree of obscuration in an 
eclipse 

Disk, is the face of the Sun or Moon as it appears to 
the eye. 

Disk of the earth, is the difference between the hori- 
zontal parallax of the Sun and Moon, and is used in the 
protraction of solar eclipses. 

Diurnal Arc, is the arc described by the celestial bodies 
from their rising to their setting. 

Diurnal Motion, is the degrees, minutes, or seconds a 
celestial body describes in twenty-four hours. 

E. 

Earth, one of the planets. Its orbit lies between Venus 
and Mars. 

Eclipse, a privation of the light of the Sun or Moon 
by the interposition of some opaque body. 

Ecliptic, the orbit T of the earth in performing her revo- 
lution round the Sun. It is inclined to the equator, 
20 



230 f ASTRONOMICAL TERMS. 

or equinoctial, at an angle of twenty-three degrees and 
twenty-eight minutes. 

Elements, in Astronomy, are those principles deduced 
from observation, by which tables of the planetary mo- 
tions are computed. 

Ephemeris Tables, containing the computations of the 
places of the heavenly bodies for every day at noon. 

Equinoctial in the heavens, or equator on the earth, 
is one of the great circles of the sphere whose poles are 
the poles of the world. 

Eccentricity, is the distance of the centre from the foci 
of the elliptical orbit of the planet. 

F. 

Fixed Stars, are such as do not appear to change their 
relative situations. 

Fomalhaut, a fixed star of the first magnitude, in the 
mouth of the southern fish. 

G. 

Galaxy, is that whitish track which appears to encom- 
pass the heavens. 

Gemini, a constellation of the Zodiac in the northern 
hemisphere, containing eighty-five stars. 

Geocentric place of a planet, is its place as seen from 
the earth. 

Geocentric Latitude of a planet, is its perpendicular 
distance from the ecliptic as seen from the earth. 

Geocentric Longitude of a planet, is its elliptical dis- 
tance from the first point of Aries as seen from the earth. 

H. 

Halo, a very conspicuous circle of about 45 degrees 
in diameter, surrounding the Sun or Moon, supposed 
to arise from the refraction of light in passing through 
the thin vapours of the atmosphere. 

Heliocentric place of a planet, is its latitude and longi- 
tude, or place in the heavens as seen from the Sun. The 
heliocentric motion of a planet is always direct, the Sun 
being its centre of motion. 



ASTRONOMICAL TERMS. 231 

Hemisphere, is the half of a globe or sphere divided by 
a plane passing through its centre. 

Hercules, a northern constellation, containing one 
hundred and thirteen stars. 

Horizon, is a great circle of the sphere, where the sight 
appears to be confined. 

Horizontal Parallax, is the parallax of a celestial body- 
when in the horizon, (see Parallax.) 

Hydra, a constellation of the southern hemisphere, 
containing 60 stars. 

I. 

Inclination, of the orbit* of a planet, is the angle which 
the plane of the planet's orbit makes with the plane of 
the ecliptic or earth's orbit. 

Ingress, is the Sun's entering any of the twelve signs 
or other parts of the ecliptic. 

J. 

Julian year, is the year established by Julius Caesar, 
now called the old style. 

Juno, one of the newly discovered planets, and the 
sixth in order from the Sun. 

Jupiter, one of the superior planets, and the largest 
in the Solar System. 

L. 

Leo, the Lion, a northern constellation of the Zodiac, 
containing ninety-five stars. 

Libra, the Balance, a constellation of the Zodiac, con- 
taining fifty-one stars. 

Limit of a planet, signifies its greatest heliocentric 
latitude. 

Longitude of a celestial body, is the distance of that 
point of the ecliptic cut by a secondary to it, passing 
through the body from the beginning of Aries. 

M. 
Magnitudes. The fixed stars, according to their size 



232 ASTRONOMICAL TERMS. 

or brightness, are divided into magnitudes. The bright^ 
est are called stars of the first magnitude ; the next in 
brightness, stars of the second magnitude ; and so on to 
the sixth or seventh magnitude ; which are the smallest 
that can be seen with the naked eye. 

Mars, a superior planet, and the fourth in order from 
the Sun. 

Mean Anomaly of a planet, is its angular distance 
from the aphelion or perihelion, supposing it to revolve 
in a circle with its mean velocity. 

Mean conjunction of the Sun and Moon, is the con- 
junction of their mean longitudes. 

Mean distance of a planet is the semi-transverse diam- 
eter of its orbit. 

Mercury, an inferior planet, the nearest to the Sun. 

Mid-heaven, called also Medum Coeli, is that point 
or degree of the ecliptic which is upon the meridian at 
any time. 

Moon, the satellite of the earth, and which is nearest 
to the earth of all the heavenly bodies. 

N. 

Nadir, is that point in the heavens directly under our 
feet. 

Nebulae, is a term applied to those telescopic stars that 
have a cloudy appearance. 

Nutation of the earth's axis, is a kind of vibratory mo- 
tion, by which its inclination to the plane of the ecliptic 
is subject to a small variation. 

O. 

Oblique Ascension, is that point of the equinoctial 
which rises with a celestial body in an oblique sphere. 

Oblique Descension, is that point of the equinoctial 
which sets with a celestial body in an oblique sphere. 

Oblique Sphere, is that position of the sphere in which 
the equator and its parallels cut the horizon obliquely. 

Observatory, a place or building fitted up with proper 
instruments for observing celestial bodies. 



ASTRONOMICAL TERMS. 233 

Occultation, is the obscuration of a star or planet by 
the interposition of the moon. 

Opposition, is that aspect of the celestial bodies when 
they are 180° asunder. 

Orbit, is the curvilinear tract, in which the planets 
perform their respective revolutions round the Sun. 

Orion, a constellation, situated upon the equinoctial, 
containing: 78 stars. 



Pallas, one of the newly discovered planets, and the 
eighth in order from the Sun. 

Parallax, is the angle formed at the centre of a star by 
two lines, one drawn from the centre, and the other from 
the surface of the earth. 

Parallax in altitude, is the difference between the true 
and apparent altitude of the body, or the difference be- 
tween the altitude at the surface and centre of the earth. 

Parallel Sphere, is that position of the sphere in which 
the equator is parallel to the horizon. 

Pegasus, a northern constellation, containing eighty- 
nine stars. 

Penumbra, is a faint shade surrounding the perfect 
shadow in an eclipse. 

Perigee, is that point of the Moon's orbit which is near- 
est to the earth. 

Perihelion, is that point of a planet's orbit which is 
nearest to the Sun. 

Perseus, a northern constellation, containing fifty-nine 
stars. 

Phases, are the different appearances of the enlighten- 
ed parts of the Moon, Venus and Mercury. 

Pisces, the last sign of the Zodiac, and which contains 
113 stars. 

Planet, is a celestial body revolving about the Sun. 
The planets may be known from the fixed stars by their 
change of situation in the heavens. 

Pleiades, or seven stars, a cluster of stars on the neck 
of the Bull. 



234 ASTRONOMICAL TERMS. 

Poles, are the extremities of the axis of the world, one 
called the north, and the other the south pole. 

Pollux, one of the twins, also the name of a star of the 
second magnitude in the constellation Gemini. 

Precession of the Equinoxes. This is a very slow 
motion of the equinoctial points in antecedentia, or from 
east to west : this motion is about fifty seconds in a year. 

Primary planets, are those that have the Sun for their 
centre of motion. ^ 

Procyon, a star of the first magnitude in the constella- 
tion Canis Minor. 

a. 

Quadrant, the fourth part of a circle, or 90 degrees. 
Quadrature, is that position of the Moon when she is 
90 degrees from the Sun. 

R 

Radius Vecta, is that imaginary line connecting the 
planet and Sun, and which, as the planet moves round 
the Sun, describes equal areas in equal spaces of time. 

Refraction, is the bending of the rays of light in pass- 
ing through the atmosphere, thereby causing the heav- 
enly bodies to appear more elevated above the horizon 
than they really are. 

Regulus, a star of the first magnitude in the constella- 
tion Leo. 

Right Ascension, is that point of the equinoctial which 
comes to the meridian with any celestial body, and is 
reckoned from the first point of Aries. 

Right Sphere, is that on which the equator and its 
parallels cut the horizon at right angles. 

S. 

Sagittarius, the Archer, one of the constellations of the 
Zodiac, containing sixty-nine stars. 

Satellites, or secondary planets, are those celestial bo- 
dies that revolve round some primary planet. 



ASTRONOMICAL TERMS. 235 

Saturn, one of the superior planets, and is the tenth 
in order from the Sun. 

Scorpio, one of the constellations of the Zodiac, con- 
taining 44 stars. 

Sirius, the dog star, one of the brightest stars in the 
heavens, and is in the constellation Canis Major. 

Solstices, is the time when the Sun enters the tropical 
points Cancer and Capricorn. 

Solstitial points, are Cancer and Capricorn. 

Spica, a star of the first magnitude in the constellation 
Virgo. 

Syzygy, means either the conjunction or opposition of 
a planet with the Sun, 

T. 

Taurus, the Bull, a sign of the Zodiac containing 141 
stars. 

Thermometer, an instrument showing the degrees of 
heat and cold. 

Tides, the periodical flux and reflux of the waters of 
the sea, supposed to be caused bv the action of the Sun 
and Moon upon the ocean. 

U. 

Umbra, the total shadow of the earth, Moon and 
planets. 

Uranus, or Herschel, a superior planet, the most re- 
mote in the solar system. 

Ursa Major, or Great Bear, a northern constellation 
containing eighty-seven stars. 

Ursa Minor, a constellation near the north pole, con- 
taining twenty-four stars, one of which is called the pole 
star. 

V. 

Yenus, one of the inferior planets. 
Vertex, is that point in the heavens directly over our 
heads called the zenith, 



236 ASTRONOMICAL TERMS. 

Vesta, one of the newly discovered planets, and the 
fifth in order from the Sun. 

Virgo, a constellation of the Zodiac containing 110 
stars. 

Z. 

Zodiac, a belt or girdle surrounding the heavens, in 
the middle of which runs the ecliptic ; it is about sixteen 
degrees in breadth. 

Zones are five Jarge divisions of the globe, viz. the 
torrid two temperate and two frigid zones. 



THE END. 



JUL 



*'■•'■■-%. 



PLATE I 



THE COPERNICAN SYSTEM 




PLATE II. 

Disk of the Sun as seen from the different planets. 




Disk of the planets compared with the Sun at 15 inches diameter. 

^ | | f $ i f 



Saturn and his Rings. 




5$ 



o||# ® 




PLATE III. 




PLATE IV. 

NEW AND FULL MOON, 




PLATE V. 




Cygrnua, the Swan. 

Fig.l. ^z 



Phenix, the Phtnix . 




Clusters of Stars: 



Nebulous Star*, 



jfy.e. 




****** « .*## -rr. i 

* Mm****** ** «^^4^ m 






-,;'_-. 



aStf*' 



^^.s^'^ 



.■■--.■ .--.-■.'■-■■.• ■ 



PLATE VIII. 

Fig. 1. tfebuleus spat in Qri&n. Fig. 2. 



Nebulm. 




Fig. 3. Double Star Epsilon Bootes. Fig. 4. Treple Star in Monoceroa. 






Fig. 5. The Constellation Orion. Fig. 6. The Constellation Canis Major. 




